Download Stable Beliefs and Conditional Probability Spaces

Stable Beliefs and Conditional Probability Spaces
MSc Thesis (Afstudeerscriptie)
written by
Konstantinos Gkikas
(born 16th of December 1988 in Athens, Greece)
under the supervision of Dr. Alexandru Baltag, and submitted to the Board of
Examiners in partial fulfillment of the requirements for the degree of
MSc in Logic
at the Universiteit van Amsterdam.
Date of the public defense: Members of the Thesis Committee:
3 September 2015
Dr. Alexandru Baltag
Prof. Dr. Johan van Benthem
Prof. Dr. Jan van Eijck
Dr. Jakub Szymanik
i
Abstract
This thesis aims to provide a conceptual framework that unifies Leitgeb’s theory of stable
beliefs, Battigalli-Siniscalchi’s notion of strong belief and the notions of core, a priori, abnormal
and conditional belief studied by Van Fraassen and Arló-Costa.
We will first present the difficulties of modeling qualitative notions of belief and belief
revision in a quantitative probabilistic setting. On one hand we have the probability 1 proposal
for belief, which seems to be materially wrong and on the other hand we have the Lockean
thesis (or any version of it) which deprives us of the logical closure of belief (Lottery Paradox).
We will argue that Leitgeb’s ([35]) theory of stability of belief is a path between this Scylla and
Charybdis.
The first goal of this thesis is to provide an extension of Leitgeb’s theory into non-classical
probability settings, where we can condition on events with measure 0. We will define the notion
of r-stable sets in Van Fraassen’s setting ([25]), using two-place functions to take conditional
probability as primitive. We will then use the notion of r-stability to provide a definition of
conditional belief similar to Leitgeb’s.
The second goal of this thesis is to develop a formal language that will express the notion of
conditional beliefs. To do so, we will first define the structures called conditional probabilistic
frames that will give us the semantics for the logic of conditional beliefs (rCBL) that we will
present.
Finally, we will use the operators (safe belief) and C (certainty) to lay the foundations
of developing a logic of stable beliefs.
ii
Acknowledgements
I would first like to thank my supervisor Alexandru Baltag for his help, patience and guidance. I deeply admire his enthusiasm and love for what he does and I am truly grateful that he
was always there, finding time to meet and work with me even in really hectic and busy periods.
I am also indebted to Prof. Dr. Johan van Benthem, Prof. Dr. Jan van Eijck and Dr.
Jakub Szymanik for taking the time to read my thesis.
Moreover, thanks goes to Paolo Galeazzi for the great project that he organized, his excellent
supervision and all the football matches he arranged during the past two years.
On a personal note, Marcel Proust wrote “Let us be grateful to the people who make us
happy; they are the charming gardeners who make our souls blossom”. In this spirit, I would
like to thank Fernando, Nigel, Julian, KH, Paolo, Rick, Pablo, Dan, Pietro, Eileen, Ana and
all the rest of my fellow logic students for filling the past two years with the most wonderful
memories. Special thanks goes to Aldo and Moo for being the best friends one could hope for.
I would also like to thank Lorena for her support throughout the past two years.
Finally, this thesis is dedicated to Christina; her smile alone was all the inspiration I needed.
Contents
Chapter 1. Introduction
1. Introductory remarks
2. Organization of the Thesis
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2
Chapter 2. Background
1. Probability 1 vs Lockean Thesis
2. Leitgeb’s Stability Theory
3. Conditioning on sets of measure 0
4. Battigalli-Siniscalchi
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5
7
9
10
Chapter 3. Conditional Probability Spaces
15
Chapter 4. r-stability
21
Chapter 5. Conditional Belief
31
Chapter 6. Probabilistic frames
35
Chapter 7. Logic of r-stable conditional beliefs
1. Syntax and Semantics
2. Axiom System
3. Soundness
4. Completeness
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42
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44
48
Chapter 8. Safe Belief, Certainty operators and the notion of quasi-stability
1. and C operators
2. quasi-stable sets
3. Conditional Belief: quasi-stability and , C operators
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Chapter 9. Conclusion
1. Overview
2. Future Work
3. Conclusion
75
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81
Bibliography
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iii
CHAPTER 1
Introduction
1. Introductory remarks
The general approach to formalizing belief in a quantitative probabilistic manner is
the probability 1 principle. According to this principle, an agent believes a proposition if
he assigns probability 1 to it. Van Fraassen argued that this principle does not allow for
distinctions to be drawn among the maximally likely propositions ([25]). Furthermore,
Leitgeb claims that this principle is materially wrong ([34, p. 1344]) since it identifies
“believing” in a proposition A with “being certain” of proposition A.
However, theories attempting to provide a quantitative probabilistic definition of belief
come face to face with the fact that drifting away from the probability 1 principle (or in
other words adapting a version of the so - called Lockean Thesis ([24])) is incompatible
with maintaining the logical closure of belief. Any theory that attempts to do so is
confronted by the lottery paradox.
In his papers [34], [35] Leitgeb develops a theory which avoids this problem by incorporating a notion of stability. He does not equate belief with probability one and at
the same time maintains its logical closure without running into the lottery paradox, as
long as rational belief is equivalent to the assignment of a stably high rational degree of
belief ([35, p. 1]). We will argue that Leitgeb’s theory gives us a more intuitive notion
of belief, in comparison to the dominant probability 1 principle. Moreover, by adopting
Leitgeb’s theory we also maintain the logical closure of belief — as Leitgeb shows the
lottery paradox can be avoided — and therefore it seems to be a win - win situation.
Leitgeb develops his theory in a classical probability setting. We will argue that classical
probabilities are unable to do belief revision. This is because conditioning on events of
measure 0 can not be defined in a classical probabilistic setting.
In this thesis, we will extend Leitgeb’s theory into conditional probability spaces,
where conditioning on events with measure 0 is allowed, having as our final goal the use
of Leitgeb’s conditional and r-stable beliefs in epistemic game theory. More specifically,
we will work in Van Fraassen’s framework as developed in [25], [3], [21], defining the
notion of r-stable beliefs in a way (almost) identical to Leitgeb. Furthermore, we will
define the structures called probabilistic frames and an operator for conditional belief.
In the introduction of his paper [18], Board argues that epistemic logic is the path
between the obscurity of Aumann structures and the complexity of belief hierarchies. He
argues that a logical approach offers simplicity and transparency. Simplicity, due to the
semantic structures used to provide truth conditions for formulas of the formal language;
structures which can be easily adapted to provide epistemic models for games. And
transparency because of the straightforward interpretation of the language and axiom
system that provide the syntax of the logic ([18, p. 53]).
1
2
1. INTRODUCTION
Following Board, we will present our logic of conditional belief. The language of this
logic will be the language of epistemic logic, augmented by adding the modal operator
Biφ ψ that will stand for “agent i’s belief in ψ conditioning on φ”. Our axiom system
will be similar to Board’s ([18]). Finally, our semantics will be given by the structures
called probabilistic models. We will prove soundness and completeness of our logic with
an interesting completeness proof in which we will represent r-stable beliefs as Grove
spheres, obtain a probabilistic model from one of Board’s structures and prove a truthpreserving lemma between Board’s semantics and ours. After this lemma, completeness
of our axioms w.r.t. Board’s semantics will give us completeness w.r.t our probabilistic
semantics.
However, the language of this logic will be unable to express statements such as:
“agent i has an r-stable belief that agent j ...”. This in turn prevents us from being able to
define notions such as Common or Mutual r-stable belief in rationality. A straightforward
introduction of an operator in order to express r-stable beliefs syntactically would not be
a good idea. This is because such an operator would not satisfy the K-axiom. Therefore,
we will introduce two other modalities that satisfy the K-axiom and express our operator
for r-stable beliefs in terms of those.
We will turn to Baltag and Smets’ safe belief modality: ([4]) and we will also define
the operator C to express certainty. However, in contrast to Baltag and Smets, our safe
belief and certainty modalities are not going to be truthful, as explained in chapter 8.
With these operators at hand, we will be able to define a notion of stability of belief,
minimally different from Leitgeb’s: this notion will be called quasi-stability. Finally, in
the conclusion of this thesis, we will propose an axiomatization of these modalities hoping
to develop a logic of certainty and safe belief in a future paper.
2. Organization of the Thesis
The thesis is organized as follows.
In chapter 2 we will give a brief presentation of all the background work. We will first
discuss Leitgeb’s stability theory of belief in a classical probability setting. Furthermore,
we will also present Battigalli-Siniscalchi’s notions of conditional and strong belief.
In chapter 3 we will develop our framework. We will work in Van Fraassen’s setting, taking conditional probability functions as primitive. Moreover, we will use Van
Fraassen’s notions of a priori, normal and abnormal sets and prove useful properties
about them.
In chapter 4 we will define our notion of r-stable beliefs. Our definition will be similar
to Leitgeb’s, but we will allow conditioning on sets with measure 0 as well. We will also
prove some important properties of r-stable sets. We will show that our r-stable sets are
well-founded w.r.t. inclusion, as long as they are not a priori and we will also show that
they are well-ordered w.r.t. a new relation we will define, the quasi-subset relation.
In chapter 5 we will define our notion of conditional belief, based on the notion
of r-stability. We will also prove that our conditional belief operator is closed under
conjunction and is consistent w.r.t. normal sets.
In chapter 6 we will define the structures called probabilistic frames. We will also
make a brief comparison of our stable beliefs with Battigalli-Siniscalchi’s strong belief.
2. ORGANIZATION OF THE THESIS
3
In chapter 7 we will present the logic of conditional belief. Along with soundness, we
will also provide a completeness proof based on Board’s belief revision structures ([18]).
In chapter 8, we will define two new operators: (safe belief) coming from Baltag
and Smets ([4]) and C (certainty). Moreover, we will define a new notion of r-stability,
that of quasi-stability. Finally, we will show that with certain restrictions it is possible
to express both quasi-stability and conditional belief in terms of and C.
Finally, in chapter 9 we will present a possible axiomatization of and C and discuss
about future work and possible applications of r-stable beliefs.
CHAPTER 2
Background
1. Probability 1 vs Lockean Thesis
It is well known that there are severe strains between probabilities and belief ([25]).
As Van Fraassen writes ([25]) “they (probability and belief) seem too intimately related
to exist as separate but equal; yet if either is taken as the more basic, the other may
suffer.” In this section of the thesis, we will explain why modeling qualitative notions of
belief and belief revision in a quantitative probabilistic setting is far from trivial.
By qualitative belief, we usually mean belief in the sense that either it is believed that
A is the case, or that ¬A is the case, or that neither of these is the case, i.e. that our
agent is agnostic w.r.t. A.
By quantitative belief, we refer to the assignment of a degree (a numerical degree, for
example a probability number) of belief to propositions, degrees which serve as measures
of the “strength” of an agent’s belief in a proposition. Typically, believing a proposition
A with a degree of 1 means that the agent is certain that A is true, while a degree of 0
means that the agent is certain that ¬A is true, i.e. that A is false [34, p. 1339].
On the qualitative side, doxastic logic usually assumes the KD45 axioms for belief:
K
D
4
5
B(φ → ψ) ⇒ (Bφ → Bψ)
¬B(φ ∧ ¬φ)
Bφ ⇒ BBφ
¬Bφ ⇒ B¬Bφ
with B being the modal operator for belief.
The K axiom implies that the agent will believe all the logical consequences of his
beliefs. Axiom D is also called “Consistency of Belief” and it tells us that the agent can
not believe the contradictory proposition. Axiom 4 is also called “Positive Introspection”
and it tells us that an agent believes his own beliefs. Finally, axiom 5 is also known
as “Negative Introspection” and it tells us that if an agent does not believe φ then he
believes that he does not believe φ.
On the quantitative side now, we have probability functions. According to the
Bayesian view degrees of belief obey axioms of the probability calculus. Therefore, we
ascribe probability numbers (as mentioned above) to sentences or formulas ([34, p. 1343]).
For that, we need a probability function P :
P : L → [0, 1]
accompanied by certain axioms and rules.
E.g. if A ∈ L is a proposition which is logically true, then P (A) = 1.
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6
2. BACKGROUND
Moreover, most writers also assume finite additivity, i.e.:
If A, B are inconsistent with each other, then:
P (A ∨ B) = P (A) + P (B)
The two most popular proposals for bridging principles relating qualitative and quantitative belief are the following:
1. Probability 1 proposal:
B(A) iff P (A) = 1
2. Lockean Thesis:
B(A) iff P (A) ≥ r, for some r ∈ ( 21 , 1).
The first principle is one of the standard ways of defining belief in epistemic game
theory ([12], [13]). According to principle 1., an agent believes the proposition A if and
only if he assigns probability 1 to it. Now using this interpretation of belief, one is able
to derive all the classical doxastic axioms presented above. However, this definition of
belief seems — as Leitgeb puts it ([34, p. 1344]) — materially wrong. Van Fraassen
argues ([25, p. 2]) that this principle “treats belief as on a par with tautologies” and
there is no distinction among the maximally likely propositions. Imagine for example
that Aldo rationally believes that his friend Moritz is going to call him. In that case,
Aldo would (probably) refrain from accepting a bet in which he would win one drachma
if he was right and lose 100.000 euros if he was wrong. However, given that he believes
this proposition to the degree of 1, he should be eager to accept such a bet. Hence, it is
possible to rationally believe a proposition with a probability less than 1 and such a case
is out of the reach of this principle.
The second principle, was called “Lockean Thesis” by Richard Foley in [24] and identifies belief with a high degree of probability for some threshold r ∈ ( 12 , 1). The big
advantage of this principle is that it presents a much more intuitive definition of belief,
allowing us to draw a distinction between being certain of A (P (A) = 1) and believing
A (P (A) > 21 ), in the sense of considering A to be more probable than its negation.
However, this approach has its own problems as well. Adopting the Lockean Thesis, one
has to give up the logical closure of belief unless r = 1. The argument is the infamous
Lottery paradox ([25], [8]). In a lottery of 1.000 tickets believed to be fair, one agent
believes with degree 0,999 that each single ticket is not the winning one. But then the
agent should not believe with degree 0,999 the conjunction of all these, i.e.: that no ticket
is the winning one.
Therefore, as Leitgeb puts it ([34, p. 1345]) the probability 1 proposal is logically fine
but materially wrong, while the Lockean Thesis seems to be materially fine but logically
wrong.
2. LEITGEB’S STABILITY THEORY
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2. Leitgeb’s Stability Theory
In his Stability Theory of Belief ([35]), Leitgeb argues that: “it might be possible
to have one’s cake and eat it too” ([35, p. 5]). Leitgeb’s formalization of belief is based
on what he calls the “Humean conception of belief” and on a modified Lockean Thesis.
He argues that according to Loeb ([36]), Hume held the following view in his Treatise
of Human Nature: “it is rational to believe a proposition in case it is rational to have
a stably high degree of belief in it” ([35, p. 33]). The key-word here (that is absent in
the Lockean thesis) is stable. What does it mean to have a stably high degree of belief
in a proposition in a quantitative probabilistic setting? Leitgeb provides the following
formalization of this view.
His stability theory of belief consists of the following three principles ([35, pp. 7,13]).
Take W a finite set of possible worlds and:
P1 There is a (uniquely determined) consistent proposition BW ⊆ W , such that for
all propositions A we have: B(A) iff BW ⊆ A.
P2 Function P is a classical probability measure over P(W ), the powerset of W ,
i.e.: for all A, B ⊆ W :
• P (W ) = 1,
• if A is inconsistent with B, then P (A ∨ B) = P (A) + P (B),
• P (B|A) = P P(B∩A)
(A)
P3 For all A ⊆ W such that BW is consistent with A and P (A) > 0 we have
P (BW |A) > 21 and if P (BW ) = 1, then BW is the least proposition A ⊆ W with
P (A) = 1.
Principle P1 tells us that there is a proposition BW that is consistent, which has the
property that all its supersets (including BW as well) are believed. The obvious question
here is: what determines BW and how does it have this special property? The next
principle P2, essentially defines a classical probability measure P . Finally in the last
principle P3, this measure P is used to specify the proposition BW . It tells us that BW
has the property that it maintains a high degree of probability, while conditioning on any
proposition A that is first of all consistent itself (meaning has a probability higher than
0) and second it is consistent with BW . Moreover, if BW has a probability equal to 1,
then it is the smallest such proposition.
What is important to note here is that principle P3 includes Leitgeb’s definition of a
P r -stable set, which is the core of his theory of belief.
Definition 2.1. Let P be a classical probability measure on P(W ) and let r ∈ [ 21 , 1)
a number. A set A ∈ P(W ) is P r -stable if P (A|B) > r, for all B ∈ P(W ) such that
A ∩ B 6= ∅ and P (B) > 0.
If we now think of P (A|B) as the degree of A under the supposition of B, then a
P r -stable proposition has the property that whatever proposition B that is consistent
with proposition A and has a probability bigger than 0 (i.e. is considered probable) is
supposed, the probability of proposition A will be bigger than 12 , i.e. bigger than its
negation. Therefore the idea here is that a proposition is stable if it is considered more
probable than its negation, whatever consistent comes its way. Moreover, it is also important to notice that a P r -stable set also has a probability higher than 12 itself, since we
can always conditionalize on W , the set of all possible worlds ([34, p. 1359]).
8
2. BACKGROUND
The following example ([34, p. 1348],[35, p. 4]) will give a better idea of P r -stability.
Example 1. Take W a set of possible worlds: W = {w1 , w2 , w3 , ..., w8 }. Take P a
classical probability measure on P(W ), such that:
P ({w1 }) = 0, 54, P ({w2 }) = 0, 342, P ({w3 }) = 0, 058, P ({w4 }) = 0, 03994, P ({w5 }) =
0, 018, P ({w6 }) = 0, 002, P ({w7 }) = 0, 00006, P ({w8 }) = 0.
Now take r = 21 .
We can think of these eight possibilities as descriptions built from 3 propositions: A, B,
C.
Take w1 to correspond to A ∧ B ∧ ¬C, w2 to A ∧ ¬B ∧ ¬C, w3 to ¬A ∧ B ∧ ¬C, w4 to
¬A ∧ ¬B ∧ ¬C, w5 to A ∧ ¬B ∧ C, w6 to ¬A ∧ ¬B ∧ C, w7 to ¬A ∧ B ∧ C and w8 to
A ∧ B ∧ C.
This probability space looks like this:
C
0, 03994
0, 002
0, 018
0, 00006
0
0, 342
A
0, 54
0, 058
B
Now take r = 3/4 to be our threshold.
Then we get the following P 3/4 − stable sets:
{w1 , ..., w5 }, {w1 , ..., w6 }, {w1 , ..., w7 }, {w1 , ..., w8 }.
Consider for example the set {w1 , ..., w5 }.
This set is P r -stable because for all Y such that Y ∩ {w1 , ..., w5 } and P (Y ) > 0 we have
that P ({w1 , ..., w5 }|Y ) > 34 .
And now we turn back to principle P1 to obtain Leitgeb’s definition of belief:
For some H ⊆ W and a classical probability measure P : P(W ) → [0, 1], we have that:
B(H) holds if and only if ∃S : P r -stable set such that S ⊆ H.
Therefore, the agent believes a proposition, if it is entailed by one of his stable beliefs.
Now Leitgeb shows that his theory solves the Lottery Paradox ([35, p. 25,26]) and he
essentially proves the soundness of the KD45 axiom system. Therefore he full-fills his
promise that there might be a way to have the cake and eat it too.
He shows that his stable beliefs have some interesting properties, e.g. that they are nested
3. CONDITIONING ON SETS OF MEASURE 0
9
and well-founded w.r.t. inclusion. This in turn entails that we can take stable beliefs as
our Grove spheres for a sphere model for belief revision, a fact that we will use to provide
one completeness proof later on in this thesis.
Leitgeb develops his theory and provides all these results in a classical probability
setting (Kolmogorov’s axioms). However, this setting is too restrictive. This is because
in a classical probability setting conditioning on events with measure 0 is not defined.
Suppose for example that we have a space W and A, B ⊆ W such that P (A) = 0.
Then P (B|A) = P P(A∩B)
, a fraction that is not defined since the denominator is 0. Now
(A)
this restriction poses an issue in belief revision, since as Halpern writes “That makes
it unclear how to proceed if an agent learns something to which she initially assigned
probability 0” ([29]). Hence when an agent is confronted with the occurrence of an event
she considered impossible, instead of revising her beliefs she would raise her hands in
despair. As Baltag and Smets write “it is well known that simple (classical) probability
measures yield problems in the context of describing an agent’s beliefs and how they can
be revised”([8, p. 3]). Although consideration of events with measure 0 might seem to
be of little interest, Halpern argues ([29, pp.1,2]) that it plays an essential role in game
theory, “particularly in the analysis of strategic reasoning in extensive form games and in
the analysis of weak dominance in normal form games”. See for example (among others)
[12], [13], [20], [16], [17], [31], [30]. Moreover, conditioning on events with measure 0 is also
crucial in the analysis of conditional statements in philosophy ([1], [37]) and in dealing
with monotonicity in AI ([33]). One way of preempting this problem without giving up
classical probability is to demand that only impossible events can have probability 0.
However, this entails that agents never have any wrong beliefs about anything, which
seems to be a rather severe constraint. Hence Baltag and Smets’ conclusion that classical
probabilities can not deal with any non-trivial belief revision ([8, p. 3]) seems indeed
correct. And this is the main idea that motivates this thesis: to provide an extension of
Leitgeb’s theory into non-classical probability spaces.
3. Conditioning on sets of measure 0
In [29] J. Halpern provides an excellent overview of the three most popular approaches
of dealing with conditioning on sets of measure 0: conditional probability spaces (CPS’s),
nonstandard probability spaces (NPS’s) and lexicographic probability systems (LPS’s).
The idea behind CPS’s goes back to Popper ([38]) and de Finetti ([22]) and essentially
is to take conditional probability as primitive. This allows for the measure µ(V |U ) to be
defined even if µ(U |W ) = 0, for W a space and U, V ⊆ W ([29, pp. 2,3]).
In nonstandard probability spaces, the idea is that there are infinitesimals that can
be used to model events that might have infinitesimally small probability but are still
10
2. BACKGROUND
observable ([29, pp. 2,3]). This idea goes back to [40] and has been used in economics
([31], [30]), in AI ([33]) and in philosophy ([1], [37]).
Finally, lexicographic probability systems were recently introduced by Blume, Brandenburger and Dekel ([16, 17]).
A lexicographic probability system is a sequence (µ0 , µ1 , µ2 , ...) of probability measures. The idea is that the first measure of this sequence µ0 is the most important one,
then µ1 is the next and so on. The probability assigned to some event E by (µ0 , µ1 ) is
µ0 (E) + µ1 (E) for some infinitesimal . Then even if µ0 (E) = 0, E still has a positive
probability given of course that µ1 (E) > 0 ([29, pp. 2,3]).
Halpern also provides some very interesting equivalence results between these three
approaches. In particular Hammond ([31]) shows that CPS’s are equivalent to a subclass
of LPS’s, called: lexicographic conditional probability spaces (LCPS’s) as long as the
state space is finite and conditioning on any nonempty set is possible ([29, pp. 2,3]).
On the other hand, Halpern shows that if the state space is finite, NPS’s are equivalent to LPS’s ([29]). For more on these results, we refer the reader to [29], [31].
In this thesis, we adopt the Popper-Renyi theory of conditional probability systems
(CPS’s) ([25], [8], [39],[26], [38]), taking conditional probability as primitive.
More specifically, we will adopt Van Fraassen’s setting as developed in [25], [3], [21].
4. Battigalli-Siniscalchi
In their papers [12], [13] Battigalli and Siniscalchi develop their notions of conditional
and strong belief using conditional probability systems. Their theory follows the probability 1 principle of quantitative representation of belief, equating “believing H given
E” with “assigning probability 1 to H when conditioning on E”. Throughout the thesis,
we will be comparing (whenever possible) our setting and theories of conditional and
r-stable beliefs with B-S’s work. In particular, we will see that there is a direct analogy
between B-S’s work and ours, since we will show that our notion of r-stability can express
B-S’s notion of strong belief. Furthermore, this comparison shows that there is a tight
connection between the theory presented in this thesis and B-S’s work, a connection that
opens directions for future applications of stable beliefs in epistemic game theory, as we
will discuss in chapter 9.
In this section, we will present B-S’s notions of conditional and strong belief as developed in [12], [13].
To do so, we will need to define a lot of notions from the field of epistemic game
theory (strategies, payoff types, type spaces, rationality and so on). The reader that is
already familiar with these notions is asked to jump directly to Definition 2.7. On the
other hand, the reader that is not acquainted with these concepts at all should be aware
that we will only give a very brief presentation and is referred to [12], [13] among others.
4. BATTIGALLI-SINISCALCHI
11
Consider a set I = {1, ...|I|} of players, a finite collection H of non-terminal histories,
including the empty history ∅ and a finite collection of terminal histories Z. Moreover,
Θi is a finite collection of payoff types for each player i ∈ I and ui : Z × Θ → R, where
Θ = Θ1 × Θ2 × ... × ΘI .
Now each element θi ∈ Θi represents player i’s information about the payoff-aspects of
the game. If the set Θ contains only one element, then the game has complete information.
At each stage of the game, all players are informed about the history that just occurred. However, no one is informed about each others payoff types. Therefore, each
player’s actions depend on previous histories that have occurred, but not on his information θi . We will denote the set of actions of player i depending on previous history h ∈ H
as: Ai (h). If there is only one active player at each h ∈ H, we say that the game has
perfect information.
For every i ∈ I, we will use Si to denote
S the set of strategies available to him. A
strategy is defined as a function si : H →
Ai (h), with si (h) ∈ Ai (h) for all h. Now
h∈H
S = Πi∈I Si and S−i = Πj6=i Sj .
For any h ∈ H ∪ Z, S(h) denotes the set of strategy profiles which induce history h.
Its projections on Si and S−i are denoted by Si (h) and S−i (h) respectively.
Now we also use Σi = Si × Θi to denote the set of strategy-payoff type pairs for player i
and we let Σ = Πi∈I Σi and Σ−i = Πj6=i Σj .
Finally, we have the notation required to define a payoff function Ui : Σi × Σ−i → R
as usually: for all z ∈ Z, (si , θi ) ∈ Σi and (s−i , θ−i ) ∈ Σ−i , if (si , s−i ) ∈ S(z), then
U (si , θi , s−i , θ−i ) = ui (z, (θj )j∈I ).
Finally, let H(si ) = {h ∈ H : si ∈ Si (h)} denote the collection of histories consistent with
si .
For a given measure space (Xi , Xi ), consider a non-empty collection Bi ⊆ Xi of events
such that ∅ ∈
/ Bi .
The collection Bi is a collection of observable events concerning x, which x ∈ Xi is the
“true” element (real world).
Definition 2.2. A conditional probability system (or CPS) on (Xi , Xi , Bi ), is a mapping µ(·|·) : Xi × Bi → [0, 1] such that, for all B, C ∈ Bi and A ∈ Xi , we have that:
• µ(B|B) = 1,
• µ(·|B) is a probability measure on (Xi , Xi ),
• A ⊆ B ⊆ C implies µ(A|B)µ(B|C) = µ(A|C).
Denote the set of conditional probability systems on (Xi , Bi ) by ∆Bi (Xi ).
Now B-S obtain what they call player i’s first order (conditional) beliefs about her opponent’s behavior and payoff types by taking Xi = Σ−i and Bi = {B ⊆ Σ−i : B =
S−i (h) × Θ−i for h ∈ H}. We denote the collection of CPSs defined on (Σ−i , Bi ) by
∆H (Σ−i ). Now to obtain player i’s higher-order beliefs, B-S introduce the notion of
an extensive-form type space ([13, p.: 361]). Each agent’s epistemic type tj ∈ Tj is
used to parametrize her conditional beliefs. Therefore, a state of the world is an array
ω = (ωj )j∈I = (sj , θj , tj )j∈I of srategies, payoff types and epistemic types. We now consider a set of possible worlds Ω = Πj∈I Ωj ⊆ Πj∈I (Σj × Tj ). Player i has conditional
12
2. BACKGROUND
beliefs about the strategies, payoff types and epistemic types of her opponents.
Hence, we specify the structure (Xi , Bi ) as: Xi = Πj6=i Ωj = Ω−i and Bi = {B ∈ Xi : B =
{(s−i , θ−i , t−i ) ∈ Ω−i : s−i ∈ S−i (h)} for h ∈ H}. We use ∆H (Ω−i ) to denote the set of
CPSs on (Ω−i , Bi ).
And now we define the structure called type space:
Definition 2.3. ([13],[14]) A type space on (H, S(·), Θ, I) is a tuple
T = (H, S(·), Θ, I, (Ωi , Ti , gi )i∈I )
such that for every i ∈ I, Ti is a compact topological space and:
• Ωi is a closed subset of Σi × Ti such that projΣi Ωi = Σi ,
• gi = (gi,h )h∈H : Ti → ∆H (Ω−i ) is a continuous mapping.
Notice that gi (ti ) = (gi,h (ti )h∈H ) is agent i’s conditional probability system. This
according to Battigalli and Siniscalchi denotes the beliefs of the epistemic type ti . Therefore, at any possible world ω = (si , θi , ti )i∈I ∈ Ω we specify player i’s strategy (si ), her
disposition to believe (gi (ti )) and her payoff type θi ([13, p.: 362]).
The natural question is whether there exists a type space that encodes all “conceivable” hierarchical beliefs. This question has been answered in the affirmative ([19]) and
B-S give the following definition of the complete-type space:
Definition 2.4. ([13]) A belief-complete type space on (H, S(·), Θ, I) is a type space
T = (H, S(·), Θ, I, (Ωi , Ti , gi )i∈I ) such that for every i ∈ I, Ωi = Σi × Ti and the function
gi maps Ti onto ∆H (Πj6=i Σj × Tj ).
B-S show in [12] that it is always possible to construct a belief-complete type space.
Now the basic behavioral assumptions in game theory is that each player i chooses
and carries out a strategy si ∈ Si that is optimal given her payoff type θi and her beliefs,
conditional upon any history consistent with si ([13, p. 363]). Here we define the notion
of best reply:
Definition 2.5. ([13, p. 363]) Fix a CPS µi ∈ ∆H (Σ−i ). A strategy si ∈ Si is a
sequential best reply to µi for payoff type θi ∈ Θi if and only if for every h ∈ H(si ) and
every s0i ∈ Si (h):
P
(Ui (si , θi , s−i , θ−i ) − Ui (s0i , θi , s−i , θ−i )) × µ({(s−i , θ−i )}|S−i (h) × Θ−i ) ≥ 0
(s−i ,θ−i )∈Σ−i
For any CPS µ ∈ ∆H (Σ−i ) let ri (µi ) denote the set of pairs (si , θi ) ∈ Σi such that si
is a sequential best reply to µi for θi .
We will also include the following notation to define rationality. Fix T a type space and
for every player i ∈ I, define:
fi = (fi,h )h∈H : Ti → [∆(Σ−i )]H
as her first-order belief mapping, i.e. for all ti ∈ Ti and h ∈ H:
fi,h (ti ) = margΣ−i gi,h (ti ).
Finally, we define rationality of a player:
4. BATTIGALLI-SINISCALCHI
13
Definition 2.6. Player i is rational at a state ω in T if and only if
ω ∈ Ri = {(s, θ, t) ∈ Ω : (si , θi ) ∈ ri (fi (ti ))}.
After all these concepts in place B-S move on to define their notion of (conditional)
probability-one belief, or (conditional)certainty as they call it ([13, p.: 364]).
So let Ai denote the σ − algebra of events E ⊆ Ω such that E = Ω−i × projΩi E.
We can see Ai as the collection of events concerning Player i. The collection of events
concerning Player i’s opponents: A−i is similarly defined.
Definition 2.7. The conditional (probability-one) belief operator for player i ∈ I
given some history h ∈ H is a mapping Bi,h : A−i → Ai , defined by:
∀E ∈ A−i : Bi,h (E) = {(s, θ, t) ∈ Ω : gi,h (ti )(projΩ−i E) = 1}.
For any E ∈ A−i , Bi,h (E) is read as: “Player i would be certain that her opponent’s
strategies, payoff and epistemic types are consistent with E were she to observe history
h”. Now B-S argue that their conditional belief operator satisfies the standard properties: Conjunction, Monotonicity and that for any E ∈ A−i , Bi,h (E) is measurable ([13,
p.: 362]).
And now for their notion of Strong Belief:
Definition 2.8. ([13, p.: 365]) For any type space T , define the operator SBi : A−i →
Ai by SBi (∅) = ∅ and:
T
SBi (E) =
Bi,h (E)
h∈H:E∩[h]6=∅
for all events E ∈ A−i − {∅} and [h] := Πj∈I Sj (h) × Θj × Tj is the event “history h
occurs”.
We say that player i strongly believes that an event E 6= ∅ is true if and only if he is
certain of E at all histories consistent with E.
These are B-S’s notions of conditional and strong belief. We will be referring to them at
certain points throughout this thesis, in order to compare them with our work.
CHAPTER 3
Conditional Probability Spaces
In this chapter, we will define the structures called conditional probability spaces.
These structures will be the core of this thesis and will be based on two-place probability
functions. We will also show that classical probability spaces can be recovered from our
conditional probability spaces.
Moreover, we will present Van Fraassen’s notions of normal, abnormal and a priori
sets ([25], [3], [21]) and prove that they have certain interesting properties.
We begin with the definition of a σ − algebra.
Definition 3.1. Let X be some set and 2X its powerset.
Then a subset Σ ⊆ 2X will be called a σ − algebra on X if it satisfies the following:
• X ∈ Σ,
• A ∈ Σ ⇒ X − A ∈ Σ ( Σ is closed under complements),
• A1 , A2 , A3 , ... ∈ Σ ⇒ A1 ∪ A2 ∪ A3 ∪ ... ∈ Σ ( Σ is closed under countable unions).
It follows that the empty set: ∅ is in Σ and also that Σ is closed under countable
intersections as well.
We proceed with the definition of a two-place probability function. As mentioned
above the idea behind CPS’s is to take conditional probability as primitive.
Let W be the set of all possible worlds, F a σ − algebra on W .
Definition 3.2. The two-place function P (·|·) : F × F → [0, 1] defined on F × F
with the following requirements:
• for any fixed set A ∈ F , the map P A : F → [0, 1] such that
P A (B) = P (B|A), ∀B ∈ F,
is either a countably additive probability measure on F, or P (B|A) has constant
value 1,
• (Multiplication Axiom) for all A, B, C ∈ F:
P (B ∩ C|A) = P (B|A)P (C|B ∩ A),
will be called a two-place probability function over F × F.
For some A ∈ F, define: P (A) = P (A|W ), with W being the whole space of possible
worlds.
And now we proceed with the definition of a conditional probability space:
Definition 3.3. We will call the structure (W, F, P ) with
15
16
3. CONDITIONAL PROBABILITY SPACES
• W a set of possible worlds,
• F a σ − algebra on W ,
• P a two-place probability function over F × F as defined above,
a conditional probability space.
Moreover, for some A ∈ F, define: Ac = W − A, i.e. Ac is the complement of A.
Observation 3.4. At this point, observe that classical probability spaces (W, F, P )
with P a unary countably additive probability measure: P : F → [0, 1], can be recovered
as a special case of conditional probability spaces.
Proof. Consider (W, F, P ) a space such that:
• W a set of possible worlds,
• F a σ − algebra on W ,
• P (·) : F → [0, 1] such that for A ∈ F, P (A) is a countably additive probability
measure and
– P (B|A) = P P(B∩A)
, if P (A) > 0,
(A)
– P (B|A) = 1, if P (A) = 0.
Now we need to establish that the Multiplication Axiom P (B∩C|A) = P (B|A)P (C|B∩A)
holds for the probability function P .
So take A, B, C ∈ F. We have the following cases:
• Case 1. P (A) = 0. Then since B ∩ A ⊆ A and P (·) is a probability measure, we
get that P (B ∩ A) = 0 as well. Therefore:
1 = P (B ∩ C|A) = P (B|A) = P (C|B ∩ A) = 1.
• Case 2. P (A) > 0. Then we have the following subcases:
– Subcase 1. P (B ∩ A) = 0. Then since B ∩ A ∩ C ⊆ B ∩ A and P (·) is a
probability measure, we get that P (B ∩ A ∩ C) = 0 as well. Thus:
0=
P (B ∩ C ∩ A)
P (B ∩ A)
= P (B ∩ C|A) = P (B|A) =
= 0.
P (A)
P (A)
– Subcase 2. P (A ∩ B) > 0. Then:
P (B ∩ C|A) =
P (B ∩ C ∩ A)
P (B ∩ C ∩ A) P (B ∩ A)
=
= P (B|A)P (C|B ∩ A).
P (A)
P (A)
P (B ∩ A)
We have now reached an essential part of Van Fraassen’s theory, the normal and
abnormal sets ([25]). Abnormal sets play the role of “absurdities”. They are the sets (or
events) that not only have measure 0 but are also considered so absurd by the agent that
if they occur, our agent is so confused that he is willing to believe anything.
Definition 3.5. Normal and abnormal sets.
• Let A ∈ F. If P A (B) is a probability measure, for B ∈ F, we will call A normal.
• Let A ∈ F. If P A (B) has constant value 1 for B ∈ F, we will call A abnormal.
It is important to notice that a normal set may have measure 0. This means that we
do not expect the agent to believe it, but he still might. For example, the rationals as
a subset of the reals have measure 0. However, it might happen that we randomly pick
3. CONDITIONAL PROBABILITY SPACES
17
a real and it turns out to be a rational. Therefore, the rationals Q are a normal set of
measure 0 ([21, p. 6]).
Definition 3.6. A conditional probability space (W, F, P ) in which W is abnormal,
will be called trivial.
Notice that if W is abnormal, then all P (A|B) = 1 for all A, B ∈ F.
Observation 3.7. For A, B ∈ F: P (A ∩ B|A) = P (B|A).
Proof. Pick A, B ∈ F. We have two cases:
• Case 1. A is normal. Then the Multiplication Axiom tells us that: P (A ∩ B|A) =
P (A|A)P (B|A ∩ A).
Now P (A|A) is a probability measure and therefore: P (A|A) = 1.
Hence P (A ∩ B|A) = P (B|A).
• Case 2. A is abnormal. Then P (A ∩ B|A) = 1 = P (B|A).
We now proceed with the definition of a priori sets. This notion is the exact opposite
of abnormality. As Van Fraassen puts it, an a priori proposition A is not epistemically
distinguishable from the tautology W (the whole space). Intuitively, this means that the
negation of A is considered absurd, or even better abnormal.
Definition 3.8. A priori and contingent sets.
• We will call a set K ∈ F a priori if and only if K c is abnormal.
• We will call a set K ∈ F contingent if and only if K is not a priori.
Note that Van Fraassen has not specified non-a priori (our contingent) sets. Though
it may be a bit too much to have normal, abnormal, a priori and contingent sets all together, we decided that such a distinction will be useful — especially in the next chapters.
Now our normal and abnormal sets have some interesting properties that will be of use
later on.
Observation 3.9. In a classical probability space (W, F, P ) (as in 3.4), the sets with
probability 0 are abnormal and the sets with probability 1 are a priori.
Proof. Take (W, F, P ) a classical conditional probability space as in 3.4.
Consider A ∈ F such that P (A) = 0. Then P (·|A) has constant value 1. This entails
by Definition 3.5 that A is abnormal.
Now consider A ∈ F such that P (A) = 1. Then since P (A) is a probability measure,
we have that P (Ac ) = 0. Hence Ac is abnormal. This entails that A is a priori (Definition
3.8).
And now we proceed with certain properties of normal and abnormal sets.
Property 3.10. If A is abnormal and B is normal, then: P (A|B) = 0
18
3. CONDITIONAL PROBABILITY SPACES
Proof. Take A, B ∈ F such that A is abnormal and B is normal.
Then P (∅|B) = P (∅|A)P (A|B), but B is normal thus P (∅|B) = 0 and A is abnormal,
thus P (∅|A) = 1, hence P (A|B) = 0.
Property 3.11. Supersets of normal sets are normal.
Proof. Pick A ∈ F a normal set. Assume towards a contradiction that ∃B ∈ F
such that A ⊆ B and B is abnormal.
Then we have that: 0 = P (∅|A) = P (B|A) = P (B ∩ A|A) = P (A|A) = 1.
Contradiction.
Property 3.12. Subsets of abnormal sets are abnormal.
Proof. Pick A ∈ F an abnormal set. Consider B ∈ F such that B ⊆ A. Then if B
is normal, the previous property entails that A is normal as well. Hence, B can only be
abnormal.
This property is also proved in [21, p.: 7].
Property 3.13. A countable union of abnormal sets is abnormal.
Proof. Consider X, Y ∈ F two abnormal sets. Assume towards a contradiction that
X ∪ Y is normal.
Then the multiplication axiom entails that: P (∅|X ∪ Y ) = P (∅|X)P (X|X ∪ Y ).
We know that P (∅|X) = 1, since X is abnormal.
Moreover, we have assumed that X ∪ Y is normal and hence P X∪Y (·) is a probability
measure.
Hence, P (∅|X ∪ Y ) = 0. This entails that P (X|X ∪ Y ) = 0.
With an analogous argument we also get that P (Y |X ∪ Y ) = 0.
However 1 = P (X ∪ Y |X ∪ Y ) = P (X|X ∪ Y ) + P (Y |X ∪ Y ) − P (X ∩ Y |X ∪ Y ) and
hence combining these we get a contradiction.
Assume now that
S Xi : i ∈ N are countably many abnormal sets in F.
Consider Z = Xi .
i
We have that Z ∈ F by the definition of a σ − algebra.
Now assume towards a contradiction that Z is normal.
We get that P (Xi |Z) = P (∅|Z) = 0.
But P Z (·) is a probability measure, thus P (Z|Z) = 0.
Contradiction.
Observation 3.14. An intersection of normal sets (either finite intersection or not)
is not necessarily normal as well.
3. CONDITIONAL PROBABILITY SPACES
19
Proof. Counterexample:
Consider W = {w1 , w2 , w3 , w4 } and consider a two-place function P as defined above
such that: P (wi |W ) = 41 , for all i ∈ {1, 2, 3, 4}.
Now take X = {w1 , w2 } and Y = {w3 , w4 }.
Then X and Y are normal sets but X ∩ Y = ∅ and hence X ∩ Y is abnormal.
Observe that Battigalli and Siniscalchi’s Conditional Probability Systems as defined
in 2.21 are similar to our two-place probability functions as defined in 3.2 along with our
normal and abnormal sets in Definition 3.5.
First of all observe that B-S’s collection Bi is essentially our collection of normal
sets. For them, Bi is a collection of observable events — or “relevant hypotheses” —
concerning the real world x. This implies that the agent considers the elements of Xi − Bi
as non-relevant hypotheses, which in our setting amounts to them being abnormal.
Moreover, B-S define their CPS µ on Xi × Bi , demanding that µ(·|B) is a probability
measure on (Xi , Xi ). Therefore, if B ∈ Bi then µ(·|B) is a probability measure. Otherwise
(i.e. if B ∈
/ Bi ), µ(·|B) is not defined.
Now in their setting, our two-place probability function P is defined on Xi × Xi ,
demanding that P (·|B) is a probability measure if B is normal and otherwise P (·|B) = 1.
Hence, if B ∈ Bi , we have that P (·|B) is defined but has constant value 1.
Finally, B-S replace our Multiplication Axiom with
A ⊆ B ⊆ C implies µ(A|B)µ(B|C) = µ(A|C)
for A ∈ Xi (i.e. A is possibly abnormal — or µ(·|A) is not defined) and B, C ∈ Bi .
Now if we in turn assume our Multiplication Axiom for a two-place probability function
P : Xi × Xi → [0, 1]:
∀A, B, C ∈ Xi : P (B ∩ A|C) = P (B|C)P (A|B ∩ C),
and take A, B, C ∈ Xi such that B, C are normal and A ⊆ B ⊆ C, we derive that
P (A|B)P (B|C) = P (A|C).
1For Battigalli and Siniscalchi, a conditional probability system (or CPS) on (X , X , B ), is a mapping
i
i
i
µ(·|·) : Xi × Bi → [0, 1] such that, for all B, C ∈ Bi and A ∈ Xi , we have that:
• µ(B|B) = 1,
• µ(·|B) is a probability measure on (Xi , Xi ),
• A ⊆ B ⊆ C implies µ(A|B)µ(B|C) = µ(A|C).
CHAPTER 4
r-stability
In this chapter we will define stable sets. We will also prove that they have some
handy properties that will be essential in the following chapters.
Our definition is similar to Leitgeb’s ([34], [35]). The idea behind the notion of an rstable proposition, is that it maintains a probability high enough (bigger than 12 ) whatever
consistent with it happens. Similarly to Leitgeb, r represents our agent’s threshold.
Throughout the whole chapter we will be talking about r-stability with the number
r ∈ ( 12 , 1] specified in the beginning. This is because we assume that an agent appears
in a game with his own threshold predetermined. This means that we assume that each
agent has already specified a number bigger than one half and smaller or equal to 1 that
serves as his own degree of gullibility/mistrust. Our assumption is that before the game
begins each agent has already picked his threshold in ( 12 , 1].
As written in the introduction Leitgeb proves that r-stable sets are nested and wellfounded w.r.t. the ⊆ relation. Sadly, we can not do the same unless we change the
definition of r-stable sets, because in our case things are more complicated than in a
classical probabilistic setting. Allowing conditioning on events with measure 0 makes it
impossible to define stable beliefs similarly to Leitgeb and at the same time maintain
nestedness w.r.t. the ⊆ relation. Assume for example that X, Y are two a priori r-stable
sets. Then, their relative complement ((X − Y ) ∪ (Y − X)) might be abnormal, but
nevertheless non-empty. Then, it is not the case that X ⊆ Y or Y ⊆ X.
There are two ways out of this situation: either change the definition of r-stable sets,
demanding that all r-stable sets are contingent, or prove some version of nestedness for
r-stable sets. We decided in favor of the second way out and defined a quasi-subset
relation: ⊆q such that X ⊆q Y iff X is included in Y , apart from an abnormal part of
it. Finally we established nestedness of r-stable sets w.r.t. ⊆q , i.e. for X, Y r-stable sets:
X ⊆q Y or Y ⊆q X.
Another issue was proving well-foundedness of r-stable beliefs w.r.t. the ⊆ relation.
The difference in our framework is that it might be the case that an r-stable set has an
abnormal subset. This would imply (as we will prove later on) that this r-stable set is
a priori. But in this case, it is not necessary that well-foundedness holds, since it might
as well be the case that we have an infinitely descending sequence of a priori sets. Once
again we could either change the definition of r-stable sets or prove a version of wellfoundedness. We could define r-stable sets by demanding that they are contingent (but
we rejected that already), or by demanding that all non-empty subsets of an r-stable set
are normal. 2 Aiming to have a definition of r-stable sets as general as possible, in order
2This
property is called Finesse by Arló-Costa ([3]) and it holds for Van Fraassen’s belief cores. The
proof can be found in [21, p.: 7].
21
22
4. R-STABILITY
for them to be straight-forwardly comparable with Leitgeb’s stable sets and B-S’s strong
beliefs, we decided to prove a version of well-foundedness, i.e. only for contingent r-stable
sets3.
Finally, we proved two important propositions that will be used extensively throughout the rest of the thesis. These propositions tell us that an r-stable set with a non-empty
abnormal subset is a priori and that if S is an a priori set and S ∩ E is non-empty and
abnormal then E is abnormal.
So we begin by considering W a set of possible worlds and (W, F, P ) a conditional
probability space as defined above.
Pick now r ∈ ( 21 , 1] our agent’s threshold.
We begin with the definition of r-stable sets.
Definition 4.1. Define the following:
• We will call a set S ∈ F r-stable set, if:
for all B ∈ F such that B ∩ F 6= ∅ : P (S|B) ≥ r.4
• We will call a set S ∈ F r-superstable set, if:
for all B ∈ F : P (S|B) ≥ r.
• We will call a set S ∈ F certain, if:
P (S|W ) = 1.
• We will call a set S ∈ F absolutely certain if:
for all B ∈ F : P (S|B) = 1 .
Observation 4.2. Consider S ∈ F:
The following are equivalent:
1. S is r-superstable
2. S is absolutely certain
3. S is 1-superstable
4. S is a priori
Proof. We will prove the following equivalences:
• 1. ⇔ 4.
(1 ⇒ 4) Assume that X ∈ F is r-superstable and that X c is normal.
Then P (X|X c ) ≥ r ⇔ P (W |X c ) − P (X c |X c ) ≥ r ⇔ 0 ≥ r.
Contradiction and thus X c is abnormal and X is a priori.
(4 ⇒ 1) A direct consequence of the definition of a priori sets.
• 2. ⇔ 4.
(2 ⇒ 4) Suppose X ∈ F is absolutely certain and assume that X c is normal.
Then P (X|X c ) = 1 ⇔ P (W |X c ) − P (X c |X c ) = 1 ⇔ 0 = 1.
Contradiction, thus X c is abnormal and therefore X is a priori.
(4 ⇒ 2) A direct consequence of the definition of a priori sets.
3Notice
that Leitgeb does something similar as well. In [34, p.: 1364] he proves that there is no
infinitely descending sequence of P r -stable sets with probability less than 1.
4Notice that for Leitgeb r ∈ [ 1 , 1) and P (S|B) > r, while for us r ∈ ( 1 , 1] and P (S|B) ≥ r.
2
2
4. R-STABILITY
23
• 3. ⇔ 2.
Obvious.
From now on, we fix a number r ∈ ( 12 , 1]. Unless otherwise specified, stable means
r-stable.
Now we will define the quasi-subset relation, in order to establish a version of nestedness for our stable sets.
Definition 4.3. For some X, Y ∈ F, define: X ⊆q Y iff ∃ an abnormal set Z such
that: X ⊆ Y ∪ Z.
Proposition 4.4. If for X, Y ∈ F we have that X ⊆ Y then X ⊆q Y .
Proof. Take X, Y ∈ F such that: X ⊆ Y .
We have that X ⊆ Y ∪ (X − Y ).
However, X − Y = ∅ and therefore X − Y is abnormal.
Thus X ⊆q Y .
And now we prove that our stable sets are nested w.r.t. ⊆q .
Property 4.5. If X, Y are stable sets then either X ⊆q Y or Y ⊆q X.
Proof. Assume that X, Y ∈ F are stable sets.
• Case 1. X ⊆ Y or Y ⊆ X. Then we get by Proposition 4.4 above that X ⊆q Y
or Y ⊆q X and our property holds.
• Case 2. Neither X ⊆ Y nor Y ⊆ X.
Consider the set: Z = (X − Y ) ∪ (Y − X). F is a σ − algebra hence: Z ∈ F.
Claim: Z is abnormal.
Proof. Assume towards a contradiction that Z is normal.
We have that Z ∩ X = X − Y 6= ∅ and Z ∩ Y = Y − X 6= ∅ since neither X ⊆ Y
nor Y ⊆ X.
The definition of r-stability entails that P (X|Z) ≥ r and P (Y |Z) ≥ r.
Hence we get: P (X|Z) + P (Y |Z) ≥ 2r > 1.
But this is equivalent to (by Observation 3.7):
P (X ∩ Z|Z) + P (Y ∩ Z|Z) ≥ 2r.
Hence: P (X − Y |Z) + P (Y − X|Z) ≥ 2r > 1.
24
4. R-STABILITY
But Z is normal and therefore P Z (·) = P (·|Z) is a probability measure.
Therefore we have that:
P (((X − Y ) ∪ (Y − X))|Z) − P (((X − Y ) ∩ (Y − X))|Z) ≥ 2r.
Hence: P (Z|Z) ≥ 2r > 1.
Contradiction.
Hence Z is abnormal. Now since we have that X ⊆ Y ∪ Z, we get that
X ⊆q Y .
Corollary 4.6. If X and Y are stable and at least one of X, Y is contingent, then
either X ⊆ Y or Y ⊆ X.
Now the previous proposition entails the following interesting observation:
Observation 4.7. If X, Y are stable sets, so is X ∩ Y .
Proof. Consider X, Y ∈ F stable sets.
Assume without loss of generality that Y ⊆q X.
Then there exists an abnormal set Z such that: Y ⊆ X ∪ Z.
However, we also have that Y = X ∪ (Y − X) and therefore Y − X is abnormal.
Now pick set E ∈ F such that X ∩ Y ∩ E 6= ∅.
Then:
P (X ∩ Y |E) = P (X|E) + P (Y |E) − P (X ∪ Y |E).
Now we have that: X ∪ Y = X ∪ (Y − X).
Therefore:
P (X ∩ Y |E) = P (X|E) + P (Y |E) − P (X|E) − P (Y − X|E) + P (X ∩ (Y − X)|E).
But Y − X is abnormal and therefore:
P (X ∩ Y |E) = P (Y |E) ≥ r.
Hence X ∩ Y is stable.
We also have the following property for unions of stable sets.
Property 4.8. A countable union of stable sets is stable.
Proof.SLet {Ki } a countable family of stable sets Ki ∈ F.
Then B = Ki is in F as well.
i
Now let E ∈ F a normal set such that B ∩ E 6= ∅.
We want to show that: P (B|E) ≥ r.
First notice that if E was abnormal, then P (B|E) ≥ r would hold trivially.
4. R-STABILITY
25
S
S
Since now B ∩ E = ( Ki ) ∩ E = (Ki ∩ E) is non-empty, there must be some i such
i
i
that Ki ∩ E 6= ∅.
Since Ki is stable, we get that: P (Ki |E) ≥ r.
But then, we also have that P (B|E) ≥ r since Ki ⊆ B and E is a normal set.
Hence B is indeed a stable set.
As mentioned above, we will show that contingent stable sets are well-founded. This
is the strongest result we can get without imposing any constraints on the definition of
stable sets.
Proposition 4.9. There is no infinitely descending chain of sets K1 ⊃ K2 ⊃ K3 ⊃ ...
such that all Ki are contingent stable. (Note that the relation ⊃ here is proper.)
Proof. Assume towards a contradiction that there exists an infinitely descending
chain of sets K1 ⊃ K2 ⊃ K3 ⊃ ... T
such that all Ki are contingent stable sets.
Now consider the set X = (K1 − Ki ).
i
Claim: X is normal.
Proof. Assume otherwise towards a contradiction. Then consider the set:
Y = (W − K1 ) ∪ X = K1c ∪ X.
Now Y ∈ F and Y ∩ K1 6= ∅. Now since K1 is stable, we get that:
1
P (K1 |Y ) ≥ r > .
2
Now by Observation 3.7: P (K1 |Y ) = P (K1 ∩ Y |Y ) = P (X|Y ) ≥ r.
Now K1c is normal (since K1 is contingent) and therefore by our properties above Y is
normal as well, as its superset.
Now we have assumed that X is abnormal and hence: P (X|Y ) = 0. But this is a
contradiction.
Now that we have established that X is normal, we go on with our proof:
We have that:
1 = P (X|X)
∞
X
=
P (Ki − Ki+1 |X)
i=1
N
−1
X
= lim
N →∞
P (Ki − Ki+1 |X)
i=1
= lim P (K1 − KN |X)
N →∞
= P (K1 |X) − lim P (KN |X)
N →∞
= 1 − lim P (KN |X)
N →∞
26
4. R-STABILITY
Now every KN is stable and KN ∩ X 6= ∅.
This is because KN − KN +1 ⊆ KN ∩ X, which is non-empty because KN +1 is strictly
included in KN .
Therefore, we have that P (KN |X) ≥ r, ∀N .
Hence: limN →∞ P (KN |X) ≥ r.
Thus, we get that 1 ≤ 1 − r from the equalities above.
Contradiction.
Corollary 4.10. The family of contingent stable sets is closed under arbitrary intersections.
Proof. Let G be a non-empty family of contingent stable sets.
Then by Corollary 4.6, we know that sets in G are totally ordered by ⊇.
By Proposition 4.9 there has to exist a smallest set in G (included in all others), otherwise
we would obtain an infinite descending chain of sets in G.
Then obviously the intersection of all the sets in G coincides with this smallest set of
G.
Now we have two important propositions.
Proposition 4.11. If S ∈ F is stable and there exists an abnormal set a ∈ F with
a 6= ∅, such that a ⊆ S, then S is a priori.
Proof. Take S an r − stable set and assume that there exists an abnormal set a ∈ F
with a 6= ∅ and a ⊆ S.
Assume towards a contradiction that S is not a priori.
Then W − S is normal.
Now take E = a ∪ (W − S). We have that S ∩ E = S ∩ (a ∪ (W − S)) = a 6= ∅ and since
S is r-stable we have that P (S|E) ≥ r.
By Observation 3.7 this is equivalent to: P (a|E) ≥ r.
Contradiction.
Corollary 4.12. Any non-empty subset of a contingent stable set is normal.
Proposition 4.13. For S ∈ F an a priori set, if for some A ∈ F we have that S ∩ A
is abnormal, then A is abnormal.
Proof. Take S ∈ F a priori set and assume that for some A ∈ F, S ∩ A is abnormal.
We have that S c is abnormal and therefore A − S is abnormal as well as its subset.
Moreover, we have that S ∩ A is abnormal as well.
But A = (S ∩ A) ∪ (A − S) and a countable union of abnormal sets is abnormal, by
Property 3.13.
Hence A is abnormal.
4. R-STABILITY
27
Propositions 4.11, 4.13 will give us a standard way to argue that a non-empty intersection of an r-stable set S with a normal set E is normal. This is because if we assume
otherwise, i.e. that ∅ =
6 S ∩ E ⊆ S is abnormal with S r-stable and E normal, then by
4.11 S is a priori and then by 4.13 E is abnormal. Contradiction. This argument will be
used extensively in what comes next.
The following Proposition is a generalization (to conditional probability spaces) and
strengthening of a result by Leitgeb on characterizing stable sets in a quantitative manner
(in the framework of classical probability). For its statement and its proof, we adopt the
following conventions:
1
= ∞, ∞ · 0 = 1, ∞ · x = ∞ for x > 0.
0
These conventions allow us to deal with the case r = 1, in such a way that we can
state and prove the result below in a uniform manner, for all thresholds r ∈ ( 12 , 1].
Proposition 4.14. Let X ∈ F be a contingent set, such that every non-empty subset
of X is normal. Then the following are equivalent:
(1) X is stable.
(2) For all sets Y, Z, T ∈ F such that ∅ 6= Y ⊆ X, Z ⊆ (W − X), Y ∪ Z ⊆ T , we
have
r
P (Z|T ).
P (Y |T ) ≥
1−r
(3) For all sets Y, Z ∈ F such that ∅ =
6 Y ⊆ X, Z ⊆ (W − X), we have
r
P (Z|Y ∪ Z).
P (Y |Y ∪ Z) ≥
1−r
Proof. (1) ⇒ (2). Let X be a contingent stable set, and Y, Z, T ∈ F be sets s.t.
∅=
6 Y ⊆ X, Z ⊆ (W − X), Y ∪ Z ⊆ T .
Since X is stable and X ∩ (Y ∪ Z) = Y 6= ∅, we must have that
P (Y |Y ∪ Z) = P (X ∩ (Y ∪ Z)|Y ∪ Z) = P (X|Y ∪ Z) ≥ r.
Since Y is a non-empty subset of X it must be normal (by the assumption of this Proposition). So Y ∪ Z is also normal, and hence (given that Y and Z are disjoint, since Y ⊆ X
and Z ⊆ (W − X)) we have that
P (Z|Y ∪ Z) = 1 − P (Y |Y ∪ Z) ≤ 1 − r.
By applying the Multiplication Axiom, we obtain that:
r
· (1 − r) · P (Y ∪ Z|T ) ≥
1−r
r
r
≥
· P (Z|Y ∪ Z) · P (Y ∪ Z|T ) =
· P (Z|T ).
1−r
1−r
(Note that, with the above conventions, this derivation works even for r = 1.)
P (Y |T ) = P (Y |Y ∪ Z) · P (Y ∪ Z|T ) ≥ r · P (Y ∪ Z|T ) =
(2) ⇒ (3). Obvious (just take T := Y ∪ Z).
(3) ⇒ (1). Suppose towards a contradiction that (3) holds for X, but that X is not
stable.
Then there exists some E ∈ F such that E ∩ X 6= ∅ but P (X|E) < r.
Take Y = E ∩ X, Z = E ∩ (W − X).
28
4. R-STABILITY
Since Y is a non-empty subset of X, it follows that Y is normal (by the assumption of
this Proposition), and hence E is normal (since Y ⊆ E). Clearly, we have
P (Y |E) = P (E ∩ X|E) = P (X|E) < r,
and from the normality of E (and since Y and Z are disjoint) we obtain that
P (Z|E) = 1 − P (Y |E) > 1 − r.
Note that the conditions of (3) are satisfied by this choice of Y and Z and that moreover
Y ∪ Z = E. By applying (3), we get that
r
r
· P (Z|E) ≥
· (1 − r) = r
P (Y |E) ≥
1−r
1−r
(where again the derivation goes through even for r = 1, given our conventions).
From this, we get:
P (X|E) = P (E ∩ X|E) = P (Y |E) ≥ r,
which contradicts the choice of E (which was s.t. P (X|E) < r).
Corollary 4.15 (Leitgeb’s characterization). Suppose we have a classical probability
space (W, F, P ) (i.e. in which all sets of measure 0 are abnormal), and let X ∈ F with
P (X) < 1. Let r ∈ ( 12 , 1). Then the following are equivalent:
(10 ): X is r-stable.
(20 ): For all sets Y, Z, T ∈ F such that ∅ =
6 Y ⊆ X, Z ⊆ (W − X), we have
r
P (Y ) ≥
· P (Z).
1−r
Proof. (10 ) ⇒ (20 ). Assume that X is r-stable.
Note that X is contingent, since it has probability different from 1, so its complement is
normal (since it has non-zero probability and the space is classical).
By Corollary 4.12 from before, we know that every non-empty subset of a contingent
stable set is normal.
Hence we can apply the implication (1) ⇒ (2) of the above Proposition (Proposition
4.14), with T := W .
(20 ) ⇒ (10 ). Assume (20 ). We will prove first the assumption of the above Proposition
4.14: let Y ∈ F s.t. ∅ =
6 Y ⊆ X.
r
Take Z := W − X. Then P (Z) = 1 − P (X) > 0. By (20 ), we have P (Y ) ≥ 1−r
· P (Z) > 0
(since r 6= 1 and P (Z) > 0).
Hence Y is normal (since we are in a classical probability space), and we have proved the
assumption of the above Proposition: every non-empty subset of X is normal.
We now prove condition (3) of the above Proposition 4.14: assume given Y, Z ∈ F
such that ∅ =
6 Y ⊆ X, Z ⊆ (W − X).
0
Using (2 ) and the fact that P (Y ∪ Z) 6= 0 (since P (Z) > 0), we obtain:
P (Y |Y ∪ Z) =
P (Y )
r
P (Z)
r
≥
·
=
· P (Z|Y ∪ Z),
P (Y ∪ Z)
1 − r P (Y ∪ Z)
1−r
i.e. condition (3) of the above Proposition.
Applying now the implication (3) ⇒ (1) of the above Proposition, we obtain that X is
r-stable.
4. R-STABILITY
29
Example 2. Consider Example 1 again: W = {w1 , ..., w8 } and P a classical probability measure on P(W ) such that:
P ({w1 }) = 0, 54, P ({w2 }) = 0, 342, P ({w3 } = 0, 058, P ({w4 }) = 0, 03994, P ({w5 }) =
0, 018, P ({w6 }) = 0, 002, P ({w7 }) = 0, 00006, P ({w8 }) = 0.
Now take r = 3/4 as our threshold and take the set X = {w1 , w2 , ..., w5 }.
Notice that then X c = {w6 , w7 , w8 }.
Now we want to check whether X is 3/4 − stable.
According to the previous Corollary we need to verify that
P (Y |W ) ≥ 35P (Z|W )
for Y ⊆ X with P (Y ) > 0 and Z ⊆ X c with P (Z) > 0.
Take Z = {w6 , w7 , w8 }.
We have that 3P (Z) = 0, 00618.
And P ({w1 }) = 0, 54 > 0, 00618, P ({w2 }) = 0, 342 > 0, 00618, P ({w3 }) = 0, 058 >
0, 00618, P ({w4 }) = 0, 03994 > 0, 00618, P ({w5 }) = 0, 018 > 0, 00618.
Therefore, according to the previous Corollary X is indeed 3/4 − stable.
Finally, we have the following definition:
Definition 4.16. An ω-stable conditional probability space (W ω , F ω , P ω ) is a conditional probability space that contains only countably many contingent stable sets.
These spaces will be used in chapter 8.
5
3=
3
4
1− 34
CHAPTER 5
Conditional Belief
In this chapter we will define the notion of conditional belief on the grounds of rstability, similarly to Leitgeb ([34, p.: 1381]). Leitgeb’s theory of belief is based on what
he calls the “Humean conception of belief”. As mentioned above, for Leitgeb a proposition H is believed given E if there is some r-stable S set such that S ∩ E 6= ∅ and
S ∩ E ⊆ H.
The intuition is that an agent would rationally believe a hypothesis H given some
evidence E if one of his r-stable beliefs is consistent with the evidence E and furthermore
H is “implied” by this r-stable belief and the evidence set. In other words, we demand
that the evidence is first consistent with our agent’s current beliefs and second that it
is rational for the agent to conclude that the hypothesis H holds after observing (conditioning on) E. However, we should also consider something else that did non exist in
Leitgeb’s setting, the abnormal sets.
As mentioned in the introduction the abnormal sets are essentially our agent’s “absurdities”. When conditioning on an abnormal set a, everything gets probability 1. Something analogous will hold for our conditional belief as well, namely: when “conditioning”
on an abnormal set, everything is believed. This captures the idea that if something the
agent considered absurd occurs, then the agent is completely confused, not knowing what
to believe and thus believing everything!
Notice that we will use the notion of conditional belief to derive full / absolute belief,
by conditioning on the whole space of possible worlds.
We are also going to prove certain essential properties of our conditional beliefs,
including closure under finite intersections, monotonicity and consistency w.r.t. normal
sets. Notice that consistency of conditional beliefs w.r.t. all sets does not hold, because
as written above everything is believed (event the empty set) when conditioning on an
abnormal set.
Finally, we connect conditional belief directly with probability: H is believed given
E implies P (H|E) ≥ r, getting one direction of the Lockean thesis, exactly as Leitgeb
does for his notion of belief.
We begin by considering (W, F, P ) a conditional probability space with W a set of possible worlds.
Once again, consider a number r ∈ ( 21 , 1] to be our agent’s threshold.
31
32
5. CONDITIONAL BELIEF
Definition 5.1. For E, H ∈ F:
B E H iff B(H|E) iff either E is abnormal or ∃S : r−stable such that S ∩ E is normal
and S ∩ E ⊆ H
B E H should be read as: “H is believed given E”. We introduced two notations for
conditional belief: B E H and B(H|E) and we will be switching to one another depending
on the context (if we have a probability measure P E (·) we will be using B(H|E) and if
we have a conditional probability P (H|E) we will be using B E H).
The definition above says that a hypothesis H is believed given some evidence E if
and only if either E is abnormal or there exists an r-stable set S of which intersection
with E is normal and is a subset of H. This definition captures the idea that a belief
should be justifiable by some other persistent (stable) belief together with the evidence.
The first question is why (in the case that E is normal) do we demand that the
intersection S ∩ E is normal and not just non-empty? This is the first time that we will
use Propositions 4.11 and 4.13. Assume that S ∩ E is non empty. Then if it is abnormal,
proposition 4.11 entails that S is a priori and proposition 4.13 entails that E is abnormal.
Contradiction. On the other hand, if S ∩ E is normal, then it can not be the empty set
(the empty set is by definition abnormal). Therefore, it appears that as long as E is
normal S ∩ E being normal and S ∩ E being non empty are equivalent.
We also have the following corollary that expresses what we discussed about abnormal
sets in the introduction of this chapter.
Corollary 5.2. For H, E ∈ F, if E is abnormal then B(H|E) holds.
Proof. Follows directly from the definition of conditional beliefs.
Definition 5.3. We will say that a hypothesis H is believed by our agent and write
B(H) iff B(H|W ) holds, with W the whole space of possible worlds.
Property 5.4. For H, E ∈ F, if E is normal and B(H|E) holds, then H is normal.
Proof. Assume otherwise, i.e. for E, H ∈ F, B(H|E) holds but H is abnormal.
Then ∃S : r-stable set such that: ∅ =
6 S ∩ E ⊆ H.
Then by the properties of normal and abnormal sets we get that S ∩ E is abnormal as
well (as a subset of an abnormal set), by Property 3.12.
Now P (S|E) ≥ r since S is r-stable and S ∩ E 6= ∅.
However we also have that: P (S|E) = P (S ∩ E|E) = 0, by Observation 3.7 and the fact
that E is normal.
Contradiction.
Another way of proving this is by using propositions 4.11 and 4.13. These would give us
that since S ∩ E is abnormal and non empty then S is a priori and then E is abnormal.
Contradiction.
5. CONDITIONAL BELIEF
33
This property tells us that abnormal sets can not be believed when conditioning on
normal sets. Or in other words, abnormal sets can be believed by the agent only when
conditioning on abnormal sets.
Property 5.5. For H, E ∈ F, if H is a priori then B(H|E).
Proof. Pick H, E ∈ F such that H is a priori. Since H is a priori we have that H c
is abnormal and also that H is 1−stable.
Now if H ∩ E = ∅, then E ⊆ H c and that would make E abnormal as well by Property
3.12, in which case we get that B(H|E) holds by Corollary 5.2.
Therefore: ∅ =
6 H ∩ E ⊆ H and thus B(H|E) holds, since H is 1−stable.
As Van Fraassen mentions ([25]) the idea of a priori is the opposite of the idea of
abnormal sets. Therefore it makes sense to expect something analogous w.r.t. conditional
belief. And we do have this analogy, since everything is believed given an abnormal set
while an a priori proposition is believed given anything.
Property 5.6 (Closure of conditional belief under finite intersections). For E, H, H 0 ∈
F if B(H|E), B(H 0 |E) hold, then B(H ∩ H 0 |E) holds as well.
Proof. Take E, H, H 0 ∈ F and assume that B(H|E) and B(H 0 |E) hold.
We have two cases:
• Case 1. E is abnormal. Then by the definition of conditional belief, we immediately get: B(H ∩ H 0 |E), as desired.
• Case 2. E is normal. This, together with B(H|E) gives us that ∃S : stable set
such that ∅ =
6 S ∩ E ⊆ H.
0
Since B(H |E) holds, we get that ∃S 0 : stable set such that ∅ =
6 S 0 ∩ E ⊆ H 0.
Now consider: S 00 = S 0 ∩ S. By Observation 4.7 S 00 is stable as well and therefore
S 00 6= ∅.
We have that S 00 ∩ E 6= ∅, since S ∩ E 6= ∅ and S 0 ∩ E 6= ∅.
Also S 00 ∩ E ⊆ H and S 00 ∩ E ⊆ H 0 , since S ∩ E ⊆ H and S 0 ∩ E ⊆ H 0 .
Therefore: ∅ =
6 S 00 ∩ E ⊆ H ∩ H 00 and hence B(H ∩ H 0 |E) holds.
Property 5.7 (Consistency of conditional belief w.r.t. normal sets). For E ∈ F, if
E is normal then ¬B(∅|E).
Proof. Assume that B(∅|E) holds for E normal set in F.
Then ∃S : r-stable set such that: ∅ =
6 S ∩ E ⊆ ∅.
But this is of course a contradiction. e
Property 5.8 (Monotonicity of conditional belief). Take A, B ∈ F such that A ⊆ B.
If B(A|E) holds, then B(B|E) holds as well.
34
5. CONDITIONAL BELIEF
Proof. We have two cases:
• Case 1. E is abnormal. Then B(B|E) holds by definition, as desired.
• Case 2. E is normal. Then ∃S : r-stable set such that S ∩ E ⊆ A.
But since A ⊆ B, then S ∩ E ⊆ B as well, entailing that B(B|E) holds as well.
Lemma 5.9. If for E, H ∈ F we have B(H|E), then P (H|E) ≥ r.
Proof. Take E, H ∈ F such that B(H|E) holds.
We have two cases:
• Case 1. E is abnormal. Then P (H|E) = 1 ≥ r.
• Case 2. E is normal. Then ∃S : r-stable set such that ∅ =
6 S ∩ E ⊆ H.
E
Hence since E is normal we have that P (·) is a probability measure and therefore
P (H|E) ≥ P (S ∩ E|E) ≥ r, since S is r-stable and S ∩ E 6= ∅.
CHAPTER 6
Probabilistic frames
In this chapter we essentially switch from the single agent case we have been considering so far to the multi agent case. We do so by defining the structures called probabilistic
frames. These structures will give us the semantics for the Logic of Conditional Belief we
will present in the next chapter.
As the name implies, a probabilistic frame is based on the notion of a conditional
probability space.
First, we consider a set of agents Ag and we define a function r that assigns to each
i ∈ Ag, i.e. to each one of these agents a number ri ∈ ( 21 , 1]. This number ri serves as
the agent’s threshold, representing her own degree of gullibility/mistrust.
Now the set of possible worlds is divided into partitions Πi , one for each of our agents
i. Moreover, we are not going to have a single two-place probability function. Instead,
we will have a two-place probability function for each one of our agents at each one of our
worlds. Or in other words, we will have a function Pi for each one of our agents that will
assign a two-place probability function Piw at every possible world. Therefore, we will be
talking about the two-place probability function of agent i at world w. This function Pi
looks similar to B-S’s function gi in [12], [13].
Furthermore, we will define an operator Bi (H|E) that will stand for “the set of possible
worlds in which agent i believes H given E”. This operator will depend on agent’s i ri stable sets w.r.t. the two-place probability function Piw .
Finally, it is also important to note that we will impose certain constraints on the
partition sets and on the two-place probability functions to ensure that the sets of the
form Bi (H|E) for H, E ∈ F are measurable.
We begin by taking W to be a set of possible worlds and F a σ − algebra on W .
Moreover, consider a set of agents Ag = {1, ..., n}.
Now for each agent i we will consider a partition Πi over W ([8]).
For each w ∈ W , we will use w(i) to denote the information cell of w for agent i
induced by the partition Πi .
Moreover, define the following relation over W induced by Πi .
35
36
6. PROBABILISTIC FRAMES
Definition 6.1. For w, v ∈ W define the relation:
w ∼i v iff w(i) = v(i).
Definition 6.2. We will say that Y ⊆ W is closed under ∼i iff
∀w, v(w ∈ Y, w ∼i v ⇒ v ∈ Y ).
Definition 6.3. The structure: M = (W, F, Πi , r, Pi ), such that:
• W is a set of possible worlds
• F a σ − algebra on W
• Πi partitions of W for i ∈ Ag, such that ∀i ∈ Ag, ∀Y ⊆ W such that Y is closed
under ∼i we have: Y ∈ F,
• r : Ag → ( 21 , 1] a function assigning a number ri ∈ ( 12 , 1] to each i ∈ Ag,
• Pi : W → (F × F → [0, 1]) a function that assigns to each world w ∈ W a twoplace probability function Piw over F × F as defined above such that the following
conditions are satisfied:
(a) (W, F, Piw ) is a non-trivial conditional probability space,
(b) W − w(i) is abnormal,
0
(c) if w0 ∈ w(i), then Piw = Piw
will be called a probabilistic frame.
The restriction we imposed on the partition sets (3rd clause) is needed to ensure that
our operator for conditional belief will be measurable.
Concerning the function Pi , the idea is that for each w ∈ W , Pi induces a non-trivial
conditional probability space (W, F, Piw ), with the two-place probability function Piw forcing all subsets of w(i)c to be abnormal. Notice that this in turn entails that the set w(i)
as well as all the sets X ∈ F such that: w(i) ⊆ X are a priori sets w.r.t. Piw . Moreover,
0
we also demanded that Piw = Piw , for all w0 ∈ w(i). This entails that agent i assigns
the same probabilities to all states in W , while at any state of the same information cell
induced by the partition. We need this constraint in order to ensure that the conditional
belief operator will behave as we desire and more precisely that it will be closed under
∼i , which will in turn entail that it is measurable, by the way we defined the partitions.
Now we will define the following operator Bi : F × F → F for conditional belief:
Definition 6.4. For E, H ∈ F define:
Bi (H|E) := {w|either Piw (∅|E) = 1 or ∃S : ri −stable set w.r.t. Piw : S∩E is normal and S∩
E ⊆ H}
Conditional belief now becomes dependent on the information that the agent possesses
about the state. The interpretation is that agent i believes H given E at all states w in
which either E is abnormal w.r.t. his probability function Piw , or he has some ri -stable
belief S w.r.t. Piw such that ∅ =
6 S ∩ E ⊆ H. Therefore, we now need to specify a state
w when talking about conditional belief: “H is believed given E by agent i at state w”.
6. PROBABILISTIC FRAMES
37
Finally we show that for two measurable sets E, H B(H|E) is also measurable.
Proposition 6.5. If E, H ∈ F then Bi (H|E) ∈ F.
Proof. Pick E, H ∈ F.
We want to show that Bi (H|E) ∈ F.
We will use the following property:
Property 6.6. Bi (H|E) is closed under ∼i .6
Proof. Pick w ∈ Bi (H|E) and v ∈ W such that w ∼i v.
I want to show that v ∈ Bi (H|E).
Since v ∈ w(i) we get that Piv = Piw , by the properties of Piw .
Therefore v ∈ Bi (H|E).
Hence Bi (H|E) is indeed closed under ∼i .
Now, the definition of the partition Πi entails that Bi (H|E) ∈ F.
Example 3. We will now present an example from game theory.
Consider the game G = (I, (Si , πi )i∈I ):
A B
C 2,2 0,0
D 0,0 1,1
with I = {R, C}, the row player R and the column player C, SR = {C, D} the strategies of player R, SC = {A, B} the strategies of player C and the payoff functions
πi : Si × S−i → R, for i ∈ I, as in the figure above.
Now consider the following belief matrix:
tA
t0A
tB
1
1
tC 0; 0 0; 12
1; 12
1
3 3
4 3
t0C 12
; 0 12
; 12 12
; 12
1
3 4
4 4
tD 12 ; 1 12 ; 12 ? 12
; 12
1
3 4
4 4
0
tD 12 ; 0 12 ; 12 12 ; 12
t0B
1
0; 12
4 3
;
12 12
4 4
;
12 12
4 4
;
12 12
with tA , t0A , tB , t0B the types of player C and tC , t0C , tD , t0D the types of player R.
Moreover, we have a function Pi : Ti → (T−i → [0, 1]), for i ∈ I, a function assigning
a classical probability measure to each type of player i over his opponent’s types.
(t )
For example PR C (tA ) = 0 and is the probability that player R’s type tC assigns to player
C’s type tA
6This
is introduced as a proposition instead of a claim because we will later refer to it separately.
38
6. PROBABILISTIC FRAMES
Now notice that the only irrational types are tC and tA .
Therefore, if we examine the state (tD , tB ), we get that:
11
(t )
PR D (rationalityC ) =
12
11
(tB )
PC (rationalityR ) =
12
(t )
with PR D (rationalityC ) being the probability that player R’s type tD assigns to player
(t )
C’s rationality (w.r.t. Definition 2.6) — and analogously for PC B .
These probabilities come up by adding the probabilities that each player’s type in the
state (tD , tB ) assigns to the other player’s rational types.
Take for example player R. As we said, the only irrational types here are tC and tA . Now
3
4
at state (tD , tB ), player R’s type tD assigns probability 12
to player C’s type t0A , 12
to
4
0
player C’s type tB and 12 to player C’s type tB . Adding these, the probability measure
(t )
11
PR D assigns probability 12
to the set {t0A , tB , t0B }, which is the set of the rational types
of agent C.
(t )
Analogously for player C’s type tB probability measure PC B .
< 1 to each other’s
Hence, we have that both players R and C assign probability 11
12
rationality at state (tD , tB ). Now we will check that this belief in rationality is 35 -stable
(t )
(t )
w.r.t. to PR D and PC B . To do that, we will use Corollary 4.15.
Let’s begin with player R.
According to this Corollary, we need to check that the probability of every subset of the
(t )
set {t0A , tB , t0B } has a probability bigger than 23 7 times the probability of tA (w.r.t. PR D ):
the only irrational type of player C.
Therefore, we have:
• For the Row player R:
3
3 (t )
3 (t )
3
(t )
PR D (t0A ) =
> PR D ({tA , t0A , tB , t0B }\{t0A , tB , t0B }) = PR D (tA ) =
12
2
2
24
(since tA is the only irrational type of player C);
(t )
(t )
likewise for PR D (tB ), PR D (t0B ).
• Analogous for player C, where the only irrational type of player R is tC .
(t )
(t )
Therefore, we have that the belief in rationality is 35 -stable w.r.t. PR D and PC B .
Finally, we would like to make some comments concerning B-S’s strong and conditional
belief. In 2.7 we have that:
Bi,h (E) = {(s, θ, t) ∈ Ω : gi,h (ti )(projΩ−i E) = 1}, for h ∈ F, E ∈ A−i .
The analogous definition of B-S’s conditional belief in our framework is:
Bi,E (H) = {w ∈ W : Piw (H|E) = 1}, for E, H ∈ F.
Moreover, in 2.8 we have that:
SBi (E) =
T
h∈H:E∩[h]6=∅
7bigger
than 32 , because
r
1−r
=
3
5
1− 35
=
15
10
=
3
2
Bi,h (E).
6. PROBABILISTIC FRAMES
39
Therefore, we have that in our framework:
T
SBi (E) =
Bi,H (E) and equivalently:
{H∈F :E∩H6=∅}
T
SBi (E) =
{w ∈ W : Piw (E|H) = 1} and equivalently:
{H∈F :E∩H6=∅}
w ∈ SBi (E) iff Piw (E|H) = 1 for all H ∈ F : E ∩ H 6= ∅ and equivalently:
“E is strongly-believed at w iff E is 1-stable w.r.t. Piw ”
Therefore, B-S’s strong belief is our notion of 1-stable belief.
Now for their notion of conditional belief, it is interesting to consider the following.
In Example 3 from before, we showed that player R’s belief in player C’s rationality at state (tD , tB ) is 53 -stable. Therefore, this means that player R believes that
player C is rational at state (tD , tB ) with our notion of conditional belief. However,
(t )
11
PR D (rationalityC ) = 12
< 1. Therefore player R does not believe that player C is
rational in B-S’s sense.
CHAPTER 7
Logic of r-stable conditional beliefs
In this chapter we introduce a formal language for the notion of conditional belief. We
present our logic of r-stable conditional beliefs, to which we will be referring as rCBL.
In section 7.1 we introduce the language and the semantics of rCBL. The language
is the standard language of epistemic logic along with a new modality Biφ ψ standing for
“agent i’s belief in ψ conditioning on φ”. The semantics are given by the structures called
probabilistic models, which are essentially the frames we defined in the previous chapter
enhanced with a valuation.
In section 7.2 we present our axiom system, which is almost the same with Board’s
axiom system in his paper [18]. Board also includes the operator Bi φ in his language
(as a primitive symbol) and the axiom he calls Triv: Bi φ ⇔ Bitrue φ. His operator Bi φ
stands for the notion of absolute belief. We decided to define absolute belief in terms
of the conditional belief operator, to simplify our language (having only one primitive
symbol) and our axiom system. Board argues ([18, p. 55]) that having absolute belief
as a primitive symbol might be useful in game theoretic applications as it could encode
the beliefs of the players prior to the game. This is a fair point, however we decided in
favor of keeping our language — and axioms — as simple as possible. Furthermore, we
included the axiom D: ¬Bi ⊥ (Consistency of Belief) which will express that our space
is not trivial in the sense of Definition 3.6.
In section 7.3 we prove the soundness of our axiom system w.r.t. our probabilistic
models. We provide a separate proof that each axiom is sound.
Finally, in section 7.4 we prove completeness of our axioms w.r.t. our probabilistic
semantics. The completeness proof proceeds in the following steps. First, we present
Board’s semantics, consisting of belief revision structures, which look like a generalized
version of Stalnaker’s structures ([18, p. 52]). We also add one more condition in his
structures, referring to our axiom D. Second, we present Board’s completeness proof,
in which we have added a final clause in the proof that the canonical model is a belief
revision structure — that the canonical model satisfies our extra condition. Finally, we
prove a truth preserving lemma between Board’s semantics and ours. We show that for
any of his belief revision structures, we can construct a probabilistic model with the same
set of worlds and same valuation and such that the same sentences of our language are
satisfied at the same worlds in the two models. Then, completeness of our axiom system
w.r.t. our probabilistic semantics follows from completeness of our axiom system w.r.t.
Board’s semantics.
41
42
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
1. Syntax and Semantics
The language of rCBL is defined as follows:
Definition 7.1. Consider a set of atomic sentences At and a set of agents: Ag =
{1, .., n}.
Our language L is a set of formulas φ of rCBL and is defined recursively:
φ ::= p|¬φ|φ ∧ φ|Biφ φ
for p ∈ At, i ∈ Ag. Call our language L.
Our language is that of epistemic logic augmented by adding the operators Biφ , for i ∈ Ag
that tell us what agent i believes after learning (observing) φ.
Moreover, we use the standard abbreviations: φ ∨ ψ for ¬(¬φ ∧ ¬ψ), φ → ψ for ¬φ ∨ ψ
and φ ↔ ψ for (φ → ψ) ∧ (ψ → φ). Furthermore define: ⊥ = p ∧ ¬p for p ∈ At and
> = ¬⊥. Finally, we introduce the following abbreviation: Bi φ := Bi> φ.
Now for our semantics, consider the following structure:
Definition 7.2. A probabilistic model M is a tuple: M = (F, || · ||), such that
F = (W, F, Πi , r, Pi ) is a probabilistic frame and || · || : L → F is a function assigning a
set of worlds to each atomic proposition.
Therefore, M is of the form:
M = (W, F, Πi , r, Pi , || · ||).
Finally, we require that for all p ∈ At, we have that ||p|| ∈ F.
Truth in probabilistic models is defined as follows:
Definition 7.3. Let M = (W, F, Πi , r, Pi , || · ||) a probabilistic model, w ∈ W , i ∈ Ag
and p ∈ At. The relation between pairs (M, w) and formulas φ ∈ L is defined as
follows:
• (M, w) p iff w ∈ ||p||,
• (M, w) ¬φ iff w ∈ ||φ||c ,
• (M, w) φ ∧ ψ iff w ∈ ||φ|| ∩ ||ψ||,
||θ||
• (M, w) Biθ φ iff w ∈ Bi ||φ||
Finally we have the following important proposition:
Proposition 7.4. || · || is a well-defined function from L to F.
Proof. We proceed by induction on the length of φ.
We have that ||p|| is well defined and that ||p|| ∈ F.
We have that if ||φ|| is well defined and ||φ|| ∈ F, then ||φ||c is well defined and ||φ||c ∈ F
by the properties of a σ − algebra.
If ||φ||, ||ψ|| are well-defined and ||φ||, ||ψ|| ∈ F we have that ||φ|| ∩ ||ψ|| is well-defined
and that ||φ|| ∩ ||ψ|| ∈ F, once again by the properties of a σ − algebra.
2. AXIOM SYSTEM
43
Assume that ||θ||, ||φ|| are well-defined and ||θ||, ||φ|| ∈ F.
||θ||
Then using Proposition 6.5 and the fact that ||θ|| is well-defined we get that Bi ||φ|| is
well-defined and also in F.
2. Axiom System
Consider i ∈ Ag and φ, ψ formulas in L.
Our axioms and inference rules are the following, along with MP:
Taut
Distr
Succ
IE(a)
IE(b)
RE
LE
PI
NI
D
true
Biφ ψ ∧ Biφ (ψ → χ) ⇒ Biφ χ
Biφ φ
Biφ ψ ⇒ (Biφ∧ψ χ ⇔ Biφ χ)
¬Biφ ¬ψ ⇒ (Biφ∧ψ χ ⇔ Biφ (ψ → χ))
from ψ infer Biφ ψ
from φ ⇔ ψ infer Biφ χ ⇔ Biψ χ
Biφ ψ ⇒ Biχ Biφ ψ
¬Biφ ψ ⇒ Biχ ¬Biφ ψ
¬Bi> ⊥
Call the axiomatic system above BRSID.
As mentioned above, we have adopted Board’s axiomatic system ([18, p. 54]). However, note that as we mentioned in the introduction of this chapter, Board also has the
following axiom Triv : Bi φ ⇔ Bitrue φ, since for him Bi φ is a primitive symbol in the
language. Moreover, our axiom D is a weakening of Board’s axiom WCon : φ ⇒ ¬Biφ ⊥
([18, p. 62]).
Now true stands for any propositional tautology, while f alse stands for ¬true. As Board
argues ([18, p. 55]) this system is close to the system K of epistemic logic, corresponding roughly to the AGM axioms of belief revision. Now Taut, Dist, RE and D are
already familiar from epistemic logic (tautologies, K, Necessitation and Consistency of
Belief). IE(a) states that an agent does not revise his beliefs if he learns something that
he already believed. IE(b) states that if the agent learns something consistent with her
original beliefs, then she adds the new information to her existing beliefs, closes under
MP and thus forms her revised beliefs ([18, p. 55]). Moreover, our axioms PI and NI
are Board’s axioms TPI and TNI ([18, p. 60]) and are the axioms for total positive and
negative introspection, stating that an agent has complete introspective access to his own
beliefs.
44
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
Finally, Distr and RE jointly correspond to the AGM axiom (K ∗ 1),8 Succ is the analogue of (K ∗ 2), IE(a) is implied by (K ∗ 7) and (K ∗ 8) in the presence of (K ∗ 5), IE(b)
corresponds to (K ∗ 7) and (K ∗ 8) and LE to (K ∗ 6). Also, in Board’s setting, IE(b) also
corresponds to (K ∗ 3) and (K ∗ 4) in the presence of Triv and Board argues that Triv
itself is implied by the AGM axioms ([18, p. 55]).
3. Soundness
In this section we will prove that each axiom of BRSID is sound.
Consider therefore a set of agents Ag = {1, ..., n} and a probabilistic model M =
(W, F, Πi , r, Pi , || · ||).
Here are the proofs that the axioms above are sound:
Distr:Biφ ψ ∧ Biφ (¬ψ ∨ χ) ⇒ Biφ χ
Proof. Pick w ∈ W .
Assume that Biφ ψ and Biφ (¬ψ ∨ χ) hold at w.
• Case 1. ||φ|| is abnormal w.r.t. Piw . Then Biφ χ holds, by Corollary 5.2.
• Case 2. ||φ|| is normal w.r.t. Piw . Then since w ∈ Biφ ψ we derive that:
∃S : ri -stable set w.r.t. Piw such that ∅ =
6 S ∩ ||φ|| ⊆ ||ψ||.
φ
Now since w ∈ Bi (¬ψ ∨ χ), we get that:
∃S 0 : ri -stable set w.r.t. Piw such that: ∅ =
6 S 0 ∩ ||φ|| ⊆ ||¬ψ|| ∪ ||χ||.
00
0
Now consider the set S = S ∩ S.
By Observation 4.7 in chapter 4, S 00 is ri -stable, w.r.t. Piw .
Now S 00 ∩ ||φ|| = S ∩ S 0 ∩ ||φ|| =
6 ∅ since S ∩ ||φ|| =
6 ∅, S 0 ∩ ||φ|| =
6 ∅ and S 00 6= ∅.
Now pick x ∈ S 00 ∩ ||φ||.
Assume that x ∈ ||¬ψ||.
Then since x ∈ S 00 ∩ ||φ||, we get that x ∈ S ∩ ||φ|| and since S ∩ ||φ|| ⊆ ||ψ||, we
get that x ∈ ||ψ||. Contradiction.
Therefore if x ∈ S 00 ∩ ||φ||, we have that x ∈
/ ||¬ψ||. However if x ∈ S 00 ∩ ||φ||,
0
0
then x ∈ S ∩ ||φ|| and S ∩ ||φ|| ⊆ ||¬ψ|| ∪ ||χ||. Hence x ∈ ||χ||. Therefore
∅=
6 S 00 ∩ ||φ|| ⊆ ||χ||.
This shows that Biφ χ holds at w and our proof is complete.
8AGM
detailed
(K ∗ 1)
(K ∗ 2)
(K ∗ 3)
(K ∗ 4)
(K ∗ 5)
(K ∗ 6)
(K ∗ 7)
(K ∗ 8)
axioms (Under the numbering system of [27], as presented in Appendix A of [18]. For a more
account look at [27, 2].)
Kφ∗ is a belief set
φ ∈ Kφ∗
Kφ∗ ⊆ Kφ+
if ¬φ ∈
/ K then Kφ+ ⊆ Kφ∗
∗
Kφ = Kf alse iff φ is logically inconsistent
if φ ↔ ψ then Kφ∗ = Kψ∗
∗
Kφ∧ψ
⊆ (Kφ∗ )+
ψ
∗
if ¬ψ ∈
/ Kφ∗ then (Kφ∗ )+
ψ ⊆ Kφ∧ψ
3. SOUNDNESS
45
Unless otherwise specified, for the rest of this section all the sets are abnormal, normal,
ri -stable, contingent or a priori w.r.t. Piw .
Succ:Biφ φ
Proof. Pick a world w ∈ W .
• Case 1. ||φ|| is abnormal. Then Biφ φ holds at w, by Corollary 5.2.
• Case 2. ||φ|| is normal. Take W to be the whole space of possible worlds.
Now W is normal and 1 − stable.
Moreover, we have that: W ∩ ||φ|| = ||φ||.
Therefore Biφ φ holds at w.
IE(a):Biφ ψ ⇒ (Biφ∧ψ χ ⇔ Biφ χ).
Proof. Pick w ∈ W and assume that Biφ ψ holds at w.
• Case 1. ||φ|| is abnormal. Then ||φ|| ∩ ||ψ|| ⊆ ||φ|| and therefore ||φ|| ∩ ||ψ|| is
abnormal as well.
Hence Biφ∧ψ χ ⇔ Biφ χ holds at w, by Corollary 5.2.
• Case 2. ||φ|| is normal. Assume that Biφ∧ψ χ holds at w.
Then by Biφ ψ, we have that ∃S : ri -stable set such that: ∅ =
6 S ∩ ||φ|| ⊆ ||ψ||.
φ∧ψ
0
Moreover, by Bi χ, we have that ∃S : ri -stable set such that ∅ =
6 S 0 ∩ ||φ|| ∩
||ψ|| ⊆ ||χ||.
Now consider the set: S 00 = S 0 ∩ S.
By Observation 4.7, S 00 is ri -stable as well.
Now take the set S 00 ∩ ||φ||.
Since S ∩ ||φ|| =
6 ∅, S 0 ∩ ||φ|| =
6 ∅ and S 00 6= ∅, we have that S 00 ∩ ||φ|| =
6 ∅.
00
Now pick x ∈ S ∩ ||φ||.
Then x ∈ S ∩ ||φ|| and therefore x ∈ ||ψ||.
Now x ∈ S 0 ∩ ||φ|| ∩ ||ψ|| as well. Therefore x ∈ ||χ||.
Hence: ∅ =
6 S 00 ∩ ||φ|| ⊆ ||χ||. Thus Biφ χ holds at w.
Now for the other direction, assume that Biφ χ holds at w.
• Case 1. ||φ|| ∩ ||ψ|| is abnormal. Then Biφ∧ψ χ holds at w as well.
• Case 2. ||φ|| ∩ ||ψ|| is normal. Then:
by Biφ ψ, we have that ∃S : ri -stable set , such that: ∅ =
6 S ∩ ||φ|| ⊆ ||ψ||.
φ
0
Also, by Bi χ, we have that ∃S : ri -stable set, such that: ∅ =
6 S 0 ∩ ||φ|| ⊆ ||χ||.
00
0
00
Once again consider the set S = S ∩ S. We know that S is ri -stable as well,
by Property 4.7.
Take the set S 00 ∩ ||φ|| ∩ ||ψ||.
Now S 00 ∩ ||φ|| is non-empty.
Pick x ∈ S 00 ∩ ||φ||.
Then x ∈ S ∩ ||φ||. Therefore x ∈ ||ψ||.
Hence S 00 ∩ ||φ|| ∩ ||ψ|| =
6 ∅.
00
Now if x ∈ S ∩ ||φ|| ∩ ||ψ||, then x ∈ S 00 ∩ ||φ||.
46
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
Therefore x ∈ S 0 ∩ ||φ||. Hence x ∈ ||χ||.
Thus ∅ =
6 S 00 ∩ ||φ|| ∩ ||ψ|| ⊆ ||χ|| and hence Biφ∧ψ χ holds at w.
IE(b):¬Biφ ¬ψ ⇒ (Biφ∧ψ χ ⇔ Biφ (¬ψ ∨ χ)).
Proof. Pick a world w ∈ W .
Assume that ¬Biφ ¬ψ holds at world w. If ||φ|| is abnormal, then ¬Biφ ¬ψ can not hold
since by Corollary 5.2 everything is believed when conditioning on abnormal sets.
Therefore assume that ||φ|| is normal and that Biφ∧ψ χ holds at w.
Now we need to consider the following cases:
• Case 1. ||ψ|| is abnormal. Then ||ψ||c is a priori. Now if ||φ|| ∩ ||ψ||c is abnormal,
then by 4.13 we have that ||φ|| is abnormal, which contradicts our assumptions.
Hence ||φ|| ∩ ||ψ||c is normal and ||ψ||c is ri -stable as a priori.
Moreover, ||φ|| ∩ ||ψ||c ⊆ ||ψ||c , hence Biφ ¬ψ holds. Contradiction.
• Case 2. ||ψ|| is normal. Then we have two more cases:
– First case: ||φ ∩ ψ|| is normal. Then since ¬Biφ ¬ψ holds at w, we get that
6 ∃S : ri -stable set such that ∅ =
6 S ∩ ||φ|| ⊆ ||¬ψ||.
Therefore, for all S : ri -stable sets, we have that if S ∩ ||φ|| is normal, then
S ∩ ||φ|| ∩ ||ψ|| =
6 ∅.
Now we also have that ∃S 0 : ri -stable set such that: ∅ 6= S 0 ∩ ||φ|| ∩ ||ψ|| ⊆
||χ||.
Now pick x ∈ S 0 ∩ ||φ||.
Then either x ∈ ||ψ|| or x ∈ ||ψ||c .
If x ∈ ||ψ|| then x ∈ ||χ||.
Hence ∅ =
6 S ∩ ||φ|| ⊆ ||¬ψ|| ∪ ||χ|| and Biφ (¬ψ ∨ χ) holds at w.
– Second case: ||φ ∩ ψ|| is abnormal.
Then (||φ|| ∩ ||ψ||)c = (||φ||c ∪ ||ψ||c ) = K is a priori and 1 − stable.
Now K ∩ ||φ|| = ||φ|| − ||ψ||. If ||φ|| − ||ψ|| = ∅, then ||φ|| ∩ ||ψ|| = ||φ|| and
hence ||φ|| is abnormal, contradicting our assumption.
Therefore ||φ|| − ||ψ|| = K ∩ ||φ|| =
6 ∅.
But now we have that ¬Biφ ¬ψ holds at w.
Hence K ∩ ||φ|| ∩ (||¬ψ||)c = K ∩ ||φ|| ∩ ||ψ|| =
6 ∅.
But this means that (||φ||∩||ψ||)c ∩(||φ||∩||ψ||) 6= ∅, which is a contradiction.
Now for the other direction assume that Biφ (¬ψ ∨ χ) holds at w.
Since ¬Biφ ¬ψ holds at w, we have that: ∀S : ri -stable sets if S ∩ ||φ|| =
6 ∅ then S ∩ ||φ|| ∩
||ψ|| =
6 ∅.
Now since Biφ (¬ψ ∨ χ) holds at w, we have that: ∃S 0 : ri -stable set such that: ∅ 6=
S 0 ∩ ||φ|| ⊆ ||¬ψ|| ∪ ||χ||.
Since S 0 ∩ ||φ|| =
6 ∅, then S 0 ∩ ||φ|| ∩ ||ψ|| =
6 ∅.
0
0
Now: S ∩ ||φ|| ∩ ||ψ|| ⊆ S ∩ ||φ|| and we have that S 0 ∩ ||φ|| ⊆ ||¬ψ|| ∪ ||χ||. Therefore:
S 0 ∩ ||φ|| ∩ ||ψ|| ⊆ ||χ||.
Hence: ∅ =
6 S 0 ∩ ||φ|| ∩ ||ψ|| ⊆ ||χ|| and Biφ∧ψ χ holds at w.
3. SOUNDNESS
47
RE: from ψ infer Biφ ψ.
Proof. Pick w ∈ W .
If ψ holds, then ||ψ|| is an 1-stable set.
• Case 1. ||φ|| is abnormal. Then Biφ φ holds at w, by Corollary 5.2.
• Case 2. ||φ|| is normal. Then ||φ|| ∩ ||ψ|| =
6 ∅.
For assume otherwise. Then ∃x ∈ ||φ|| such that x ∈
/ ||ψ||.
But ψ holds everywhere in W and therefore this can not be the case.
Therefore: ∅ =
6 ||φ|| ∩ ||ψ|| ⊆ ||ψ||.
And hence Biφ φ holds at w.
LE: from φ ⇔ ψ infer Biφ χ ⇔ Biψ χ.
Proof. Pick w ∈ W .
Assume that φ ⇔ ψ holds.
Assume that Biφ χ holds at w.
• Case 1. ||φ|| is abnormal. Then so is ||ψ|| and hence Biψ χ holds at w, by Corollary
5.2.
• Case 2. ||φ|| is normal. Then ∃S : ri -stable set such that: ∅ =
6 S ∩ ||φ|| ⊆ ||χ||.
But since φ ⇔ ψ holds, then ||φ|| = ||ψ||.
Therefore ∅ =
6 S ∩ ||ψ|| ⊆ ||χ|| and Biψ χ holds at w.
Assume that Biψ χ holds at w.
Analogous.
PI:Biφ ψ ⇒ Biχ Biφ ψ
Proof. Pick w ∈ W .
Assume that Biφ ψ holds in w.
• Case 1. ||χ|| is abnormal. Then Biχ Biφ ψ holds at w, by Corollary 5.2.
• Case 2. ||χ|| is normal.
Consider the set w(i), which is 1 − stable.
Then ∅ =
6 w(i) ∩ ||χ|| ⊆ ||Biφ ψ||, since Biφ ψ holds at w.
Therefore Biχ Biφ ψ holds at w.
NI:¬Biφ ψ ⇒ Biχ ¬Biφ ψ
Proof. Pick w ∈ W .
Assume that ¬Biφ ψ holds in w.
48
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
• Case 1. ||χ|| is abnormal. Then Biχ ¬Biφ ψ holds, by Corollary 5.2.
• Case 2. ||χ|| is normal. Then consider the set w(i) which is 1 − stable.
Then ∅ =
6 w(i) ∩ ||χ|| ⊆ ||¬Biφ ψ||, since ¬Biφ ψ holds at w.
Therefore Biχ ¬Biφ ψ holds at w.
D:¬Bi> ⊥
Proof. Follows directly from the condition of probabilistic frames that (W, F, Piw )
is a non-trivial conditional probability space.
4. Completeness
In this section we prove that BRSID is a complete axiomatization w.r.t. our probabilistic models.
The idea is to prove a truth preserving lemma that will connect our probabilistic models
with Board’s belief revision structures and use Board’s completeness result.
So we will begin by presenting Board’s semantics ([18]).
Consider a set of agents: Ag = {1, ..., n} and a set of atomic propositions At.
Definition 7.5. The structure
M = hW, 4, || · ||i, such that:
• W a set of possible worlds,
• 4 a vector of binary relations over W ,
• || · || is a valuation function, assigning sets of worlds to each atomic proposition
will be called a belief revision structure.
4w
i is the plausibility ordering of agent i at world w.
w
x 4i y denotes that world y is at least as possible as world x for agent i while at world
w.
Definition 7.6. Define: Wiw := {x|y 4w
i x for some y}, the set of all the conceivable
worlds for agent i at world w.
We assume moreover that:
w
R1 ∀i, w :4w
i is complete and transitive on Wi
R2 ∀i, w :4w
i is well-founded
4. COMPLETENESS
49
R3 ∀i, w, x, y, z : if x ∈ Wiw , then: y 4xi z iff y 4w
i z
R4 ∀i, w : Wiw 6= ∅
R1 ensures that each plausibility ordering divides all the worlds into ordered equivalence
classes. Note that the inconceivable worlds, i.e. those not in Wiw , are a class unto themselves and are to be considered least plausible ([18, p. 56]).
R2 ensures that our relation is well founded.
R3 says that an agent has the same plausibility ordering in every world that is conceivable to her ([18, p. 61]).
R4 says that for every agent i and every world w, there exists some conceivable world.
Definition 7.7. For w ∈ W, X ⊆ W, i ∈ Ag, define:
w
bestw
i (X) := {x ∈ X|∀y ∈ X : y 4i x}.
So for some set X ⊆ W , i ∈ Ag and w ∈ W , bestw
i (X) is the set of the most plausible
worlds for agent i at state w.
Now for the definition of relation between pairs (M, w) and formulas of L:
Definition 7.8. Take M a belief revision structure.
For w ∈ W and i ∈ Ag, p an atomic proposition:
• (M, w) p iff w ∈ ||p||
• (M, w) φ ∧ ψ iff (M, w) φ and (M, w) ψ
• (M, w) ¬φ iff (M, w) 2 φ
w
• (M, w) Biφ ψ iff (M, x) ψ, ∀x ∈ bestw
i (||φ|| ∩ Wi ).
No surprises here, the first three clauses are straightforward and the truth conditions
for the conditional belief operator are as usual in plausibility models.
Proposition 7.9. BRSID is a sound and complete axiomatization w.r.t. finite belief
revision structures satisfying R1, R2, R3, R4.
Proof. Board proves that BRSI, which is BRSID without our last axiom D is a
sound and complete axiomatization w.r.t. the set of finite belief revision structures that
satisfy R1, R2, R3 ([18, p. 72-77]). This means that we need to prove the soundness
and completeness of the axiom D w.r.t. finite belief revision structures that satisfy R1,
R2, R3 and the extra property R4 that we added. Let’s call this class of belief revision
structures M.
For the soundness proofs check [18, p. 72-77]. Notice that the soundness of the axiom
D w.r.t. M is trivial due to our semantic assumption that Wiw is non-empty (R4).
Now for the completeness proof, we decided to present Board’s proof, along with the
part of the proof that we added concerning the axiom D and the property R4 that we
added.
50
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
So we will begin the proof with a round of definitions.
Definition 7.10. We say that a formula φ is BRSID-consistent if ¬φ is not provable
in BRSID.
Definition 7.11. A finite set of formulas {φ1 , ..., φk } is BRSID-consistent exactly
if φ1 ∧ ... ∧ φk is BRSID-consistent and an infinite set of formulas is BRSID-consistent
if all its finite subsets are BRSID-consistent.
Definition 7.12. Given two sets of formulas S ⊆ T ⊆ L we say that S is a maximal
BRSID-consistent subset of T if:
• it is BRSID-consistent and
• for all φ ∈ T but not in S the set S ∪ {φ} is not BRSID-consistent
Now to prove completeness, Board argues ([18, p. 73]) that it suffices to show that
(?) Every BRSID-consistent formula in L is satisfiable w.r.t. M
For if (?) holds and φ is a valid formula in L, then if φ is not provable in BRSID then
neither is ¬¬φ, so by definition ¬φ is BRSID-consistent. Then by (?) ¬φ is satisfiable
w.r.t. M, contradicting the validity of φ w.r.t. M.
Now we proceed with another round of definitions.
Definition 7.13. Let φ be a formula in L. Then define Sub(φ) to be the set of all
the subformulas of φ, i.e.: ψ ∈ Sub(φ) if:
• ψ = φ, or
0
• φ is of the form ¬φ0 , φ0 ∧ φ00 , Biφ φ00 and ψ ∈ Sub(φ0 ) or ψ ∈ Sub(φ00 )
Definition 7.14. Let Sub+ (φ) consist of all the formulas in Sub(φ) and their negations and conjunctions, i.e. Sub+ φ is the smallest set such that:
• if ψ ∈ Sub(φ), then ψ ∈ Sub+ (φ),
• if ψ, χ ∈ Sub+ (φ), then ¬ψ, ψ ∧ χ ∈ Sub+ (φ)
Definition 7.15. Let Sub++ (φ) consist of
• of all formulas of Sub+ (φ), together with all formulas of the form Biχ ψ, where
ψ, χ ∈ Sub+ (φ) and
• if ξ ∈ Sub+ (φ) and Biχ ψ ∈ Sub++ (φ), then Biξ Biχ ψ ∈ Sub++ (φ) and Biξ ¬Biχ ψ ∈
Sub++ (φ)
++
(φ) and their
Definition 7.16. Let Sub++
neg (φ) consist of all the formulas in Sub
negation.
4. COMPLETENESS
51
Finally, let Con(φ) be the set of maximal BRSID-consistent subsets of Sub++
neg (φ).
++
We know that every BRSID-consistent subset of Subneg (φ) can be extended to an element of Con(φ) ([15, p. 199]).
Finally, any S member of Con(φ) must satisfy the following properties:
• for every ψ ∈ Sub++ (φ), either ψ or ¬ψ is in S,
• if ψ ∧ χ ∈ S then ψ ∈ S and χ ∈ S,
• if ψ ∨ χ ∈ S then ψ ∈ S or χ ∈ S,
• if ψ ∈ S and ψ → χ ∈ S then χ ∈ S,
• if ψ ↔ χ then ψ ∈ S iff χ ∈ S,
• if ψ ∈ Sub++
neg (φ) and BRSID ` ψ then ψ ∈ S.
Finally, we introduce the notation: S/Biφ = {ψ|Biφ ψ ∈ S} to refer to the set of
formulas believed by agent i when learning that φ.
Now we proceed by constructing the canonical model.
We use maximally BRSID-consistent sets as building blocks for the canonical model:
Definition 7.17. Define the following structure:
Mφ = hW, 4, || · ||i where
• W = {ws |S ∈ Con(φ)},
S
wT if there is some ψ ∈ Sub+ (φ) ∩ T ∩ U such that S/Biψ ⊆ T ,
• wU 4w
i
• ||p|| = {wS ∈ W |p ∈ S}
The canonical model Mφ has a world wS that corresponds to every S ∈ Con(φ).
The canonical relation tells us that wT is at least as possible as wU for agent i at
world wS , if there is some formula ψ ∈ Sub+ (φ) ∩ T ∩ U such that the set of the formulas
that agent i believes at world wS when learning that ψ are a subset of T .
Now in order to prove (?) we will prove the following lemma (truth lemma).
Lemma 7.18. We have that:
For every ψ ∈ Sub(φ): (Mφ , wS ) ψ if and only if ψ ∈ S.
Now suppose that we have proved this lemma. Then we have that if φ is a BRSIDconsistent formula, then it is contained in some set S ∈ Con(φ). Then by the truth
lemma above we have that (Mφ , wS ) φ. Now if we also establish that Mφ ∈ M then we
will get that φ is satisfiable w.r.t. M as required. And this (proving that Mφ ∈ M) will
be the next important step of the proof. But for now, we will prove the truth lemma.
Proof. As always, we proceed by induction on the structure of formulas.
The Induction Hypothesis is that the truth lemma holds for all subformulas ψ ∈
Sub(φ) and we show that it holds for ψ.
We will only deal with the case that ψ is of the form Biχ ζ.
52
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
(⇐)
Assume therefore that Biχ ζ ∈ S. We want to show that (Mφ , wS ) Biχ ζ.
wS
wS
wS
S
Consider the set bestw
i (||χ|| ∩ Wi ) and assume that wT ∈ besti (||χ|| ∩ Wi ), since
χ
wS
wS
otherwise if besti (||χ|| ∩ Wi ) = ∅ then (Mφ , wS ) Bi ζ.
wS
wS
S
Since wT ∈ bestw
wT for all wU ∈ {||χ|| ∩ WiwS }.
i (||χ|| ∩ Wi ) we get that wU 4i
wS
Then by the definition of 4i from the canonical model (Definition 7.17), we get that
∃ξ ∈ Sub+ (φ) ∩ T such that S/Biξ ⊆ T .
Now we want to show that ζ ∈ T to use the Induction Hypothesis and get that
S
(Mφ , wT ) ζ. Then, we would have that the most plausible worlds w.r.t. 4w
that
i
χ
satisfies χ also satisfies ζ. Hence, we will have that (Mφ , wS ) Bi ζ and establish the
right-to-left direction. So lets proceed.
We have that S/Biξ ⊆ T . This means that S/Biξ is a BRSID-consistent set, since
otherwise T would be inconsistent, but T ∈ Con(φ) and Con(φ) is the set of maximal
BRSID-consistent subsets of Sub++
neg (φ).
Lemma 7.19. S/Biχ is a BRSID-consistent set as well.
Proof. Suppose not towards a contradiction. Then there is a finite set of formulas
{φ1 , φ2 , ..., φk } ⊆ S/Biχ such that
BRSID ` ¬(φ1 ∧ ... ∧ φk ).
Now Board shows that ([18, p. 74])
BRSID ` (Biχ φ1 ∧ ... ∧ Biχ φk ∧ ¬Biξ ¬χ) → (Biξ φ1 ∧ ... ∧ Biξ φk ).
Since wT ∈ ||χ||, i.e. (Mφ , wT ) χ, using the induction hypothesis we get that χ ∈ T .
Now Biξ ¬χ ∈ Sub++ (φ). If Biξ ¬χ ∈ S, then since S/Biξ ⊆ T , we get that ¬χ ∈ T . But T
is a member of Con(φ) and hence this contradicts that χ ∈ T . Therefore Biξ ¬χ ∈
/ S and
ξ
thus ¬Bi ¬χ ∈ S by the properties of the members of Con(φ).
Claim: Biξ φ1 , ..., Biξ φk ∈ S.
Proof. Since {φ1 , φ2 , ..., φk } ⊆ S/Biχ , we have that Biχ φ1 , ..., Biχ φk ∈ S (remember
that S/Biχ = {ψ|Biχ ψ ∈ S}).
Moreover, we just argued that ¬Biξ ¬χ ∈ S.
Hence we get that Biχ φ1 ∧ ... ∧ Biχ φk ∧ ¬Biξ ¬χ ∈ S.
Then since
BRSID ` (Biχ φ1 ∧ ... ∧ Biχ φk ∧ ¬Biξ ¬χ) → (Biξ φ1 ∧ ... ∧ Biξ φk )
holds, using the properties of the members of Con(φ), we get that
(Biξ φ1 ∧ ... ∧ Biξ φk ) ∈ S.
But this entails that Biξ φ1 , ..., Biξ φk ∈ S, once again by the properties of the members of
Con(φ).
Therefore Biξ φ1 , ..., Biξ φk ∈ S. Hence {φ1 , φ2 , ..., φk } ⊆ S/Biξ , therefore S/Biξ is not a
BRSID-consistent set. Contradiction.
4. COMPLETENESS
53
Now that we have established that S/Biχ is a BRSID-consistent set, consider its
maximal extension U .
S
By (Succ) we get that Biχ χ ∈ S. Hence χ ∈ (S/Biχ ) ⊆ U . Therefore wT 4w
wU .
i
wS
wS
wS
On the other hand, we have that wT ∈ besti (||χ|| ∩ Wi ). Hence wU 4i wT . Thus
∃λ ∈ Sub+ (φ) ∩ T ∩ U such that S/Biλ ⊆ T (Definition 7.17).
Now λ ∈ U and S/Biχ ⊆ U ⇔ {ψ|Biχ ψ ∈ S} ⊆ U . Now if Biχ ¬λ ∈ S, then
¬λ ∈ S/Biχ , thus ¬λ ∈ U . Contradiction. Therefore ¬Biχ ¬λ ∈ S.
Moreover, χ ∈ T and S/Biλ ⊆ T . Hence with the same argument ¬Biλ ¬χ ∈ S.
Furthermore, Board establishes that ([18, p. 75])
BRSID ` (¬Biχ ¬λ ∧ Biχ ζ ∧ ¬Biλ ¬χ) → (Biλ ζ ∨ Biλ (χ → ζ)).
Now since χ, ζ, λ ∈ Sub+ (φ), by the definition of Sub++ (φ), we have that Biλ ζ, Biλ (χ →
ζ) ∈ Sub++ (φ).
Moreover, we have assumed that Biχ ζ ∈ S.
Therefore, we get that (Biλ ζ ∨ Biλ (χ → ζ)) ∈ S.
Now we also have that S/Biλ ⊆ T . Hence we get that ζ ∈ T or that (χ → ζ) ∈ T .
However, if (χ → ζ) ∈ T , then combined with χ ∈ T we derive that ζ ∈ T .
wS
S
Therefore ∀wT ∈ bestw
i (||χ||∩Wi ) we have ζ ∈ T and using the induction hypothesis
we get that (Mφ , wT ) ζ, which in turn entails that (Mφ , wS ) Biχ ζ.
This ends the (⇐) direction of the truth lemma.
(⇒)
Assume that (Mφ , wS ) Biχ ζ.
Claim: The set (S/Biχ ) ∪ {¬ζ} is not BRSID-consistent.
Proof. Suppose otherwise towards a contradiction. Then it would have a maximal
BRSID-consistent extension T . By (Succ) we get that Biχ χ ∈ S. Therefore χ ∈
(S/Biχ ) ⊆ T .
S
S
Hence this means that wU 4w
wT for all U such that χ ∈ U , thus wT ∈ bestw
i
i (||χ|| ∩
WiwS ).
Now ¬ζ ∈ T (since T is a maximal BRSID-consistent extension of (S/Biχ ) ∪ {¬ζ}) and
using the induction hypothesis, we get that (Mφ , wT ) ¬ζ. Therefore, we have a world
wS
S
wT in bestw
i (||χ|| ∩ Wi ) such that (Mφ , wT ) ¬ζ.
This entails that (Mφ , wT ) ¬Biχ ζ. Contradiction.
Now since (S/Biχ ) ∪ {¬ζ} is not BRSID-consistent, there must be some finite subset
{φ1 , ..., φk , ¬ζ} that is not BRSID-consistent.
Now
BRSID ` φ1 → (φ2 → (... → (φk → ζ)...)).
By applying (RE), (Distr), (MP) we derive that: ([23])
BRSID ` Biχ φ1 → (Biχ φ2 → (... → (Biχ φk → Biχ ζ)...))
54
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
and therefore
(Biχ φ1 → (Biχ φ2 → (... → (Biχ φk → Biχ ζ)...))) ∈ S.
Now φ1 , φ2 , ..., φk ∈ (S/Biχ ), thus Biχ φ1 , ..., Biχ φk ∈ S (since S/Biχ = {ψ|Biχ ψ ∈ S}).
Therefore Biχ ζ ∈ S.
And this ends the proof of the (⇒) direction and the proof of the truth lemma.
Hence, we have established that if φ is a BRSID-consistent formula contained in
some set S ∈ Con(φ), then (Mφ , wS ) φ.
The next and final step of the proof is to show that Mφ ∈ M. That way, we will have
that every BRSID-consistent formula is satisfiable w.r.t. M, thus we will prove (?) and
in turn completeness w.r.t. M.
Proposition 7.20. Mφ ∈ M
S
Proof. We will show that 4w
is complete and transitive on WiwS , well-founded,
i
absolute and that WiwS 6= ∅.9
So let us begin.
S
• 4w
is complete on WiwS .
i
Proof. Pick wT , wU ∈ WiwS .
S
S
We w.t.s. that either wT 4w
wU or wU 4w
wT .
i
i
wS
w
From the definition of Wi and 4i , we have that there is some ψ ∈ Sub+ (φ)∩
T , such that S/Biψ ⊆ T and some χ ∈ Sub+ (φ) ∩ U such that S/Biχ ⊆ U .
Now S is a maximal BRSID-consistent set.
This entails that either Biψ∨χ ¬ψ ∈ S, or that ¬Biφ∨χ ¬ψ ∈ S.
– Case 1. Biψ∨χ ¬ψ ∈ S.
By (Succ)
Biψ∨χ (ψ ∨ χ) ∈ S. (1)
By (Distr)
((Biψ∨χ ¬ψ ∧ Biψ∨χ (ψ ∨ χ)) → Biψ∨χ χ) ∈ S. (2)
By (IE(a))
(ψ∨χ)∧χ
(Biψ∨χ χ → (Bi
ζ ↔ Biψ∨χ ζ)) ∈ S. (3)
By (Taut)
(((ψ ∨ χ) ∧ χ) ↔ χ) ∈ S. (4)
Therefore by (4) and (LE)
(ψ∨χ)∧χ
Bi
ζ ↔ Biχ ζ ∈ S. (5)
Now by (2), (1) and the fact that Biψ∨χ ¬ψ ∈ S, we get that Biψ∨χ χ ∈ S.
9This
is our addition: R4.
4. COMPLETENESS
55
(ψ∨χ)∧χ
Hence (3) gives us that Bi
ζ ∈ S if and only if Biψ∨χ ζ ∈ S, i.e.:
(ψ∨χ)∧χ
S/Bi
= S/Biψ∨χ .
(ψ∨χ)∧χ
Now (5) tells us that S/Bi
= S/Biχ .
Combining the last two sentences we get that S/Biχ = S/Biψ∨χ ⊆ U (since
we argued above that S/Biχ ⊆ U ).
S
However ψ ∨ χ ∈ T ∩ U , since ψ ∈ T and χ ∈ U , thus wT 4w
wU .
i
ψ∨χ
– Case 2. ¬Bi ¬ψ ∈ S.
By (IE(b))
(ψ∨χ)∧ψ
(¬Biψ∨χ ¬ψ → (Bi
ζ ↔ Biψ∨χ (χ → ζ))) ∈ S. (6)
By (RE), (Dist), (Taut), (MP)
(Biψ∨χ ζ → Biψ∨χ (χ → ζ)) ∈ S. (7)
Now (7) entails that if Biψ∨χ ζ ∈ S, then Biψ∨χ (χ → ζ) ∈ S.
(ψ∨χ)∧ψ
And in that case since ¬Biψ∨χ ¬ψ ∈ S, (6) gives us that Bi
ζ ∈ S.
(ψ∨χ)∧ψ
ψ∨χ
ψ
Thus S/Bi
⊆ S/Bi
= S/Bi ⊆ T .
S
But ψ ∨ χ ∈ T ∩ U as before and thus wU 4w
wT .
i
wS
S
Hence completeness of 4w
on
W
has
been
established.
i
i
S
• 4w
is transitive on WiwS
i
S
S
wU and wU 4w
wT . We
Proof. Let wT , wU , wV ∈ WiwS such that wV 4w
i
i
wS
want to show that wV 4i wT .
χ
+
S
By the definition of 4w
i , we have that ∃χ ∈ Sub (φ)∩U ∩V such that S/Bi ⊆ U
and ∃ψ ∈ Sub+ (φ) ∩ T ∩ U such that S/Biψ ⊆ T .
Now S is a maximal BRSID-consistent set.
Hence either Biψ∨χ ¬ψ ∈ S or ¬Biψ∨χ ¬ψ ∈ S.
– Case 1. Biψ∨χ ¬ψ ∈ S.
S
By the first part of the proof that 4w
is complete, we have that S/Biχ =
i
S/Biψ∨χ ⊆ U .
But ψ ∈ U contradicting that Biψ∨χ ¬ψ ∈ S.
– Case 2. ¬Biψ∨χ ¬ψ ∈ S.
ψ∨χ
S
By the second part of the proof that 4w
⊆
i is complete, we have that S/Bi
S/Biψ ⊆ T .
S
But ψ ∨ χ ∈ T ∩ U ∩ V , therefore wV 4w
wT .
i
S
• 4w
is well-founded.
i
Proof. We will show that W is finite. We will do so, by showing that
Con(φ) is finite.
Now Sub(φ) is a finite set therefore it has a finite number of maximal BRSIDconsistent subsets.
Claim: Each maximal BRSID-consistent subset of Sub(φ), has a unique
extension to a maximal BRSID-consistent subset of Sub+ (φ).
56
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
Proof. Let S be a maximal BRSID-consistent subset of Sub(φ) and S + a
maximal BRSID-consistent subset of Sub+ (φ) such that S ⊆ S + . Then ψ ∈ S +
only if either ψ ∈ S, or ψ is of the form ¬χ and χ ∈
/ S + , or ψ is of the form χ ∧ ζ
+
and χ, ζ ∈ S .
Now if there is no maximal BRSID-consistent subset S of Sub(φ) such that
S ⊆ S + , S + can not be a maximal BRSID-consistent subset of Sub+ (φ). Hence,
there is a one-to-one correspondence between the maximal BRSID-consistent
subsets of Sub(φ) and the maximal BRSID-consistent subsets of Sub+ (φ). Now there are at most 2||Sub(φ)|| logically distinct formulas in Sub+ (φ). If
++
is a maximal BRSID-consistent subset of Sub++
BRSID ` ψ ↔ χ and Sneg
neg (φ),
χ
ψ
++
++
then Bi ζ ∈ Sneg iff Bi ζ ∈ Sneg . Moreover, for every formula of the form
++
++
, by (PI) and (NI).
iff Biξ · · · Biχ ψ ∈ Sneg
Biζ Biξ · · · Biχ ψ ∈ Sneg
Hence, each BRSID-consistent subset of Sub+ (φ) has a finite number of
extensions to maximal BRSID-consistent subsets of Sub++
neg (φ). Hence Con(φ)
is a finite set. Thus W is finite and well-foundedness follows.
S
• 4w
is absolute.
i
Proof. Take wT ∈ WiwS . Then ∃ψ ∈ Sub+ (φ) ∩ T such that S/Biψ ⊆ T .
T
S
wU .
wU iff wV 4w
We must show that wV 4w
i
i
(⇒)
S
Assume wV 4w
wU . Therefore we get that ∃χ ∈ Sub+ (φ) ∩ U ∩ V such that
i
χ
S/Bi ⊆ U .
Claim: T /Biχ ⊆ S/Biχ
/ S. Then ¬Biχ ζ ∈ S.
Proof. Suppose that Biχ ζ ∈
By (NI)
(¬Biχ ζ → Biψ ¬Biχ ζ) ∈ S.
Hence Biψ ¬Biχ ζ ∈ S. But S/Biψ ⊆ T . Hence ¬Biχ ζ ∈ T . Therefore Biχ ζ ∈
/ T.
χ
χ
Hence indeed T /Bi ⊆ S/Bi ⊆ U .
T
Thus wV 4w
wU .
i
(⇐)
T
Assume wV 4w
wU . Then there is some χ ∈ Sub+ (φ) ∩ U ∩ V such that
i
χ
T /Bi ⊆ U .
Claim: S/Biχ ⊆ T /Biχ
4. COMPLETENESS
57
Proof. Suppose that Biχ ζ ∈ S.
By (PI)
(Biχ ζ → Biψ Biχ ζ) ∈ S.
Hence Biψ Biχ ζ ∈ S. But S/Biψ ⊆ T , thus Biχ ζ ∈ T . Therefore S/Biχ ⊆
T /Biχ ⊆ U .
S
Thus wV 4w
wU .
i
• For all wS ∈ Mφ we have that: WiwS 6= ∅.
Proof. We begin with the following claim:
Claim: The set S/Bi> is BRSID-consistent.
Proof. Assume not towards a contradiction. Then ∃{φ1 , ...φk } ⊆ S/Bi>
such that
BRSID ` ((φ1 ∧ ... ∧ φk ) → ⊥). (10 )
Now since φ1 , ..., φk ∈ S/Bi> , we get that Bi> φ1 , ..., Bi> φk ∈ S. Therefore
(Bi> φ1 ∧ ... ∧ Bi> φk ) ∈ S. (20 )
We also have that
BRSID ` ((Bi> φ1 ∧ ... ∧ Bi> φk ) → Bi> (φ1 ∧ ... ∧ φk )). (30 )
By (20 ), (30 ) and the fact that S is a member of Con(φ) we get that
Bi> (φ1 ∧ ... ∧ φk ) ∈ S. (40 )
By (10 ) and (RE)
(Bi> ((φ1 ∧ ... ∧ φk ) → ⊥)) ∈ S. (50 )
By (Distr)
(Bi> (φ1 ∧ ... ∧ φk ) ∧ Bi> ((φ1 ∧ ... ∧ φk ) → ⊥) → Bi> ⊥) ∈ S. (60 )
By (40 ), (50 ), (60 )
Bi> ⊥ ∈ S. (70 )
But this ((70 )) contradicts axiom (D).
Hence we have proved the claim.
Therefore, the set S/Bi> can be extended to a maximal BRSID-consistent
subset U of Sub++
neg (φ).
S
S
Now the definition of 4w
from the canonical model entails that wU 4w
wU and
i
i
wS
hence Wi 6= ∅.
Hence we have shown that WiwS 6= ∅.
Hence, we have established that (Mφ , wS ) ψ iff ψ ∈ S and that Mφ ∈ M.
Now we go back to what we set out to prove:
(?) Every BRSID-consistent formula in L is satisfiable w.r.t. M
58
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
Suppose now that φ is BRSID-consistent. Then it is contained in some set S ∈
Con(φ). Then it follows from our truth lemma that (Mφ , wS ) φ. Since we proved that
Mφ ∈ M we get that φ is satisfiable w.r.t. M. Therefore this proves (?) and we have
shown that BRSID is a complete axiomatization w.r.t. M.
Now we move on to the next step.
The completeness proof w.r.t. our probabilistic semantics, will be a combination of
the previous (7.9) and the following (7.21) proposition.
Proposition 7.21. For every finite belief revision structure M = hW, 4, || · ||i satisfying R1, R2, R3, R4, there exists a finite (non-trivial) probabilistic model M 0 , having
the same set of worlds and same valuation as M and such that the same sentences of our
language are satisfied at the same worlds in the two models, i.e.:
∀w ∈ W : (M, w) φ iff (M 0 , w) φ.
Proof. We begin by fixing some threshold r such that r ∈ ( 21 , 1).
We will prove that our logic is complete for non-trivial probabilistic models in which all
agents’ thresholds are equal to r. This in turn implies completeness for arbitrary nontrivial probabilistic models.
Suppose now that M = hW, 4, || · ||i is a finite belief revision structure, satisfying R1,
R2, R3, R4.
For w, v ∈ W , we put:
w ∼i v iff Wiw = Wiv .
This gives us the equivalence classes w(i) and the partitions Πi .
Now define the following family of subsets of w(i):
10
G w,i := {S ⊆ w(i)|S 6= ∅ and ∀x ∈ S∀y(x 4w
i y ⇒ y ∈ S)}.
Lemma 7.22. The family G w,i is nested.
Proof. Pick S 0 , S 00 ∈ G w,i .
Assume towards a contradiction that S 0 6⊆ S 00 and that S 00 6⊆ S 0 .
Now pick x ∈ S 0 − S 00 and y ∈ S 00 − S 0 .
w
w
Now since 4w
i is complete on w(i), we get that either x 4i y or y 4i x.
w
0
0
If x 4i y then since x ∈ S we get that y ∈ S , by the definition of G w,i which is a
contradiction.
If y 4w
i x we get a contradiction with an analogous argument.
10G w,i
depends on both i and w
4. COMPLETENESS
59
Therefore, we have that G w,i is a family of nested and well-founded spheres that are
upwards-closed w.r.t. the plausibility relation 4w
i .
w,i 11
Consider now S0w,i ⊂ S1w,i ⊂ ... ⊂ Sn(w,i)
to be an enumeration of all those spheres
w,i
in G .
Definition 7.23. Define N0w,i := |S0w,i | to be the cardinality of the sphere S0w,i and
w,i
define Njw,i := |Sjw,i | − |Sj−1
|, ∀j = 1, ..., n(w, i).
Now, we will proceed with the following steps:
Step 1 We will define the numbers xw,i
for j = 0, ..., n(w, i) that we will later assign
j
to all the elements of the spheres in G w,i . In particular, we will later assign
(when we will define our probability function) the same probability xw,i
to all
0
w,i
w,i
the worlds in S0 and similarly the same probability xj to all the worlds in
w,i
Sjw,i − Sj−1
for all j = 1, ..., n(w, i). This is the key idea behind the construction
of our probabilistic model.
P
r
w,i w,i
xm ), for all j ∈ {0, ..., n(w, i) − 1}.
Nm
Step 2 We will show that xw,i
j ≥ 1−r (1 −
0≤m≤j
This combined with Corollary 4.15 will later entail that all Sjw,i in G w,i are rstable sets w.r.t. the probability function we will define.
Step 3 We will define the probability function Piw .
w,i
Step 4 We will show (using Step 1) that every set S 0 ⊇ Sn(w,i)
is an a priori set w.r.t.
w,i
w,i
w,i
w
⊆
Pi (with Sn(w,i) ⊆ w(i) the largest sphere in G ). And of course, since Sn(w,i)
w
w(i), we will have that w(i) and all A ⊃ w(i) will be a priori sets w.r.t. Pi as
well. Moreover, all A ⊆ W − w(i) will be abnormal w.r.t. Piw .
Step 5 We will prove that the family of r-stable sets w.r.t. Piw is the union of G w,i and
of the a priori sets. This will be done by first using Step 2 and Corollary 4.15 to
show that all the spheres in G w,i are r-stable.
Step 6 We will define our non-trivial probabilistic model M 0 .
Step 7 Finally, we will prove the truth preserving lemma:
∀w ∈ W : (M, w) φ iff (M 0 , w) φ.
So let’s begin.
Step 1
Define the following numbers:
k=
r
.
1−r
And:
• For j = 0, ..., n(w, i) − 1, we put:
k
xw,i
for every agent i.
j = Π
(1+kN w,i )
0≤m≤j
11Notice
m
that n depends on w, i, since we may have a different number of spheres for different agents
and different worlds w. This is why we use n(w, i).
60
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
• For j = n(w, i), we put:
n(w,i)−1
P
1−
xw,i
n(w,i)
=
Njw,i xw,i
j
j=0
w,i
Nn(w,i)
for every agent i.
Now we will prove the following lemmas:
Lemma 7.24. For all j = 0, ..., n(w, i) − 1 we have that xw,i
j ∈ [0, 1]
Proof.
k
w,i
Π0≤m≤j (1 + kNm
)
k
=
w,i
(1 + kN0 )(1 + kN1w,i ) · · · (1 + kNjw,i )
k
= 1
w,i
k( k + N0 )(1 + kN1w,i ) · · · (1 + kNjw,i )
1
= 1
w,i
( k + N0 )(1 + kN1w,i ) · · · (1 + kNjw,i )
xw,i
j =
Now since r ∈ ( 12 , 1), we have that k ∈ (1, ∞).
Therefore xw,i
j ∈ (0, 1), for all j = 0, ..., n(w, i) − 1.
Lemma 7.25.
n(w,i)
P
Njw,i xw,i
j = 1
j=0
Proof.
n(w,i)
X
n(w,i)−1
Njw,i xw,i
j
X
=
j=0
w,i
w,i
Njw,i xw,i
j + Nn(w,i) xn(w,i)
j=0
n(w,i)−1
n(w,i)−1
=
X
j=0
Njw,i xw,i
j
+1−
X
Njw,i xw,i
j
j=0
=1
These numbers that we defined here will be the probabilities of the elements of the
spheres of G w,i . In Step 3, we will define our probability function in a way that all the
and similarly for all j = 1, ..., n(w, i) that
worlds in S0w,i have the same probability xw,i
0
w,i
w,i
all the worlds in Sj − Sj−1 have the same probability xw,i
j .
Step 2
We will now prove the following Lemma that will be used later to establish that the sets
in G w,i are r-stable.
4. COMPLETENESS
61
Lemma 7.26. For all j ∈ {0, ..., n(w, i) − 1}:
X
r
w,i w,i
(1 −
Nm
xm ).
xw,i
j ≥
1−r
0≤m≤j
Proof. By Induction.
Base cases:
For j = 0:
xw,i
0 =
=
k
1 + kN0w,i
1+
r
1−r
r
N w,i
1−r 0
r
1 − r + rN0w,i
r
=
(1 − N0w,i xw,i
0 )
1−r
=
Therefore indeed we have that: xw,i
0 ≥
r
(1
1−r
− N0w,i xw,i
0 ).
Now we will do one more case, for j = 1:
xw,i
1 =
=
k
(1 +
kN0w,i )(1
+ kN1w,i )
k(1+kN0w,i )−k2 N0w,i
1+kN0w,i
1 + kN1w,i
k2 N0w,i
1+kN0w,i
+ kN1w,i
k−
=
=
1
r
1−r
−
rN0w,i k
(1−r)(1+kN0w,i )
rN1w,i
1−r
w,i
k
r − rN0 1+kN
w,i
0
1 − r + rN1w,i
r − rN0w,i xw,i
0
1 − r + rN1w,i
r − rN0w,i xw,i
0
w,i w,i
r − rN0 x0
1+
=
=
(1 − r + rN1w,i )xw,i
1 =
w,i
w,i
=
(1 − r)xw,i
1 + x1 rN1
xw,i
1 =
Therefore indeed xw,i
1 ≥
r
(1
1−r
r
w,i w,i
(1 − N0w,i xw,i
0 − N1 x1 )
1−r
w,i w,i
− N0w,i xw,i
0 − N1 x1 ).
Induction Hypothesis: For j = 1, ..., n(w, i) − 2:
w,i w,i
w,i w,i
w,i w,i
r
xw,i
j = 1−r (1 − N0 x0 − N1 x1 − · · · − Nj xj ).
Inductive Step: We will show that:
62
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
w,i w,i
w,i w,i
w,i w,i
r
xw,i
j+1 = 1−r (1 − N0 x0 − N1 x1 − · · · − Nj xj+1 ).
We have that:
k
xw,i
j+1 =
w,i
w,i
(1 + kN0 )(1 + kN1w,i ) · · · (1 + kNj+1
)
k(1+kN0w,i )(1+kN1w,i )···(1+kNjw,i )−k2 N0w,i (1+kN1w,i )(1+kN2w,i )···(1+kNjw,i )−···−k2 Njw,i
(1+kN0w,i )(1+kN1w,i )···(1+kNjw,i )
xw,i
j+1 =
1+
k−
xw,i
j+1 =
xw,i
j+1 =
k2 N0w,i
1+kN0w,i
−
k2 N1w,i
w,i
(1+kN0 )(1+kN1w,i )
12
w,i
kNj+1
−···−
k2 Njw,i
(1+kN0w,i )(1+kN1w,i )···(1+kNjw,i )
w,i
1 + kNj+1
w,i w,i
w,i w,i
k − kN0w,i xw,i
0 − kN1 x1 − · · ·kNj xj
w,i
1 + kNj+1
w,i
w,i w,i
w,i w,i
xw,i
j+1 (1 − r + rNj+1 ) = r − rN0 x0 · · · −rNj xj
r
w,i w,i
w,i w,i
w,i w,i
w,i w,i
xw,i
(1 − N0w,i xw,i
0 − N1 x1 − N2 x2 − · · · − Nj xj − Nj+1 xj+1 )
j+1 =
r−1
Therefore indeed
w,i w,i
w,i w,i
w,i w,i
w,i w,i
r
≥ r−1
(1 − N0w,i xw,i
0 − N1 x1 − N2 x2 − · · · − Nj xj − Nj+1 xj+1 ).
xw,i
j+1
Step 3
Now we are going to define the probability function Piw .
First define the function µw
i : W → [0, 1] such that:

w,i

if v ∈ S0w,i
x 0
w,i
µw
xw,i
if v ∈ Sjw,i − Sj−1
, with j ≥ 1
i (v) :=
j

0
else
And now define the classical probability function Piw : F → [0, 1]:
P
Piw (A) := {µw
i (v)|v ∈ A}
12Here
we use that:
k(1 + kN0w,i )(1 + kN1w,i ) · · · (1 + kNjw,i ) − k 2 N0w,i (1 + kN1w,i )(1 + kN2w,i ) · · · (1 + kNjw,i ) − k 2 N1w,i (1 +
kN2w,i )(1 + kN3w,i ) · · · (1 + kNjw,i ) − · · · − k 2 Njw,i =
(k + k 2 N0w,i )(1 + kN1w,i ) · · · (1 + kNjw,i ) − k 2 N0w,i (1 + kN1w,i )(1 + kN2w,i ) · · · (1 + kNjw,i ) − k 2 N1w,i (1 +
kN2w,i )(1 + kN3w,i ) · · · (1 + kNjw,i ) − · · · − k 2 Njw,i
k(1 + kN1w,i ) · · · (1 + kNjw,i ) + k 2 N0w,i (1 + kN1w,i ) · · · (1 + kNjw,i ) − k 2 N0w,i (1 + kN1w,i )(1 + kN2w,i ) · · ·
(1 + kNjw,i ) − k 2 N1w,i (1 + kN2w,i )(1 + kN3w,i ) · · · (1 + kNjw,i ) − · · · − k 2 Njw,i =
k(1 + kN1w,i ) · · · (1 + kNjw,i ) − k 2 N1w,i (1 + kN2w,i )(1 + kN3w,i ) · · · (1 + kNjw,i ) − · · · − k 2 Njw,i =
(k + k 2 N1w,i )(1 + kN2w,i )(1 + kN3w,i ) · · · (1 + kNjw,i ) − k 2 N1w,i (1 + kN2w,i )(1 + kN3w,i ) · · · (1 + kNjw,i ) − · ·
· − k 2 Njw,i =
k(1 + kN2w,i )(1 + kN3w,i ) · · · (1 + kNjw,i ) + k 2 N1w,i (1 + kN2w,i )(1 + kN3w,i ) · · · (1 + kNjw,i ) − k 2 N1w,i (1 +
kN2w,i )(1 + kN3w,i ) · · · (1 + kNjw,i ) − · · · − k 2 Njw,i =
k(1 + kN2w,i )(1 + kN3w,i ) · · · (1 + kNjw,i ) − k 2 N2w,i (1 + kN3w,i ) · · · (1 + kNjw,i ) − · · · − k 2 Njw,i = k
4. COMPLETENESS
63
Now for A, B ∈ F, conditional probabilities will be given by:
P w (A∩B)
• Piw (A|B) = iP w (B) , if Piw (B) > 0 and
i
• Piw (A|B) = 1 if Piw (B) = 0.
Step 4
w,i
We will show (using Step 1) that every set S 0 ⊇ Sn(w,i)
is an a priori set w.r.t. Piw (with
w,i
Sn(w,i)
⊆ w(i) the largest sphere in G w,i ).
w,i
Lemma 7.27. Every X ⊆ W such that X ⊇ Sn(w,i)
is a priori w.r.t. Piw .
w,i
Proof. Pick X ⊆ W such that X ⊇ Sn(w,i)
.
w,i
w,i
c
Then X ⊆ W − Sn(w,i) . However, everything in W − Sn(w,i)
has probability 0 w.r.t. Piw
(Lemma 7.25) and since (W, F, Piw ) is a classical probability space (Step 3), Observation
3.9 entails that X c is abnormal, meaning that X is a priori w.r.t. Piw .
Step 5
We will first prove that the sets in G w,i are r-stable.
Lemma 7.28. Each Sjw,i is r-stable w.r.t. Piw , ∀j ∈ {1, ...n(w, i)}.
Proof. Proved directly by combining Lemma 7.26 (Step 2), Corollary 4.15 and
w,i
Lemma 7.27 for Sn(w,i)
.
Now we will show that the family of r-stable sets w.r.t. Piw is the union of G w,i and
of the a priori sets.
Lemma 7.29. The only r-stable sets w.r.t. Piw are the ones in G w,i or the a priori
sets.
Proof. Fix i and w, and suppose that K is r-stable w.r.t. Piw , but is not in G w,i or
a priori.
Hence, K is a contingent stable set. By Corollary 4.12, it follows that every non-empty
subset of K is normal.
w,i
w,i
This implies that K ∩ (W − Sn(w,i)
) = ∅ (since all subsets of W − Sn(w,i)
are abnormal, so
w,i
the intersection K ∩ (W − Sn(w,i) ) is an abnormal subset of K, hence it must be empty).
w,i
Thus K ⊆ Sn(w,i)
.
Let now j be the smallest index such that K ⊆ Sjw,i . (We know that j’s with this
property exist, since already n(w, i) has this property.)
w,i
This means that K ⊆ Sjw,i , but that K 6⊆ Sm
for any m < j. But both K and all these
spheres (having m < n(w, i)) are contingent stable sets, hence they must be comparable
by inclusion.
64
7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
Therefore, we must have that
S
w,i
w,i
⊆ K.
⊆ K for all m < j, i.e. m<j Sm
Sm
S
w,i
w,i
Moreover, this union m<j Sm
is either equal to Sj−1
(in case that j > 0) or equal to ∅
(in case j = 0), so in both cases K differs from this union (since K is stable thus nonw,i
empty, and not covered by case 1, so not equal to Sj−1
). So we have a strict inclusion
S
w,i
m<j Sm ⊂ K.
S
S
w,i
w,i
Let now w be some world w ∈ K − ( m<j Sm
) ⊆ Sjw,i − ( m<j Sm
).
w,i
It is easy to see (by our construction of the probabilities for Sj ), that we have Piw (w) =
xw,i
j .
By the same construction, we have Piw (v) = xw,i
for all worlds v ∈ Sjw,i − K ⊆ Sjw,i −
j
S
S
w,i
w,i
) received probability xw,i
) (since all the worlds in Sjw,i − ( m<j Sm
( m<j Sm
j ).
w,i
Finally, note that the inclusion K ⊆ Sj is also strict (since K is not in G w,i ). So, if
we put NK := |Sjw,i − K|, then we have NK ≥ 1.
Putting all these together, we obtain:
Piw (K|{w}
∪
(Sjw,i
xw,i
xw,i
P ({w})
1
j
j
− K)) =
= w,i
w,i
w,i ≤ w,i
w,i = ,
2
P ({w} ∪ (Sj − K))
xj + NK · xj
xj + xj
which contradicts the r-stability of K.
Step 6
Now define the following structure:
M 0 = (W, F, Πi , r, Pi , || · ||) where:
• W as in the belief revision structure M ,
• F = P(W ),
• Πi partitions of W for i ∈ Ag, given by the ∼i defined above (w ∼i v iff Wiw =
Wiv ),
• r assigns the same number r ∈ ( 12 , 1) to all i ∈ Ag,
• the conditional probabilities Piw (A|B) is given by applying the usual formula for
conditional probabilities to the above-defined classical probabilities Piw (which
are defined in terms of µw
i ):
Piw (A∩B)
w
Pi (A|B) := P w (B) ,
i
• || · || is the same as in the belief revision structure M
We have that F is a σ − algebra on W . Moreover, for some i ∈ Ag and Y ⊆ W
such that Y is closed under ∼i , we have that Y ∈ F, by the definition of ∼i . Finally,
for each i ∈ Ag and w ∈ W we have that Piw is a two-place probability function over
F × F (Observation 3.4) such that the space (W, F, Piw ) is not trivial, the set W − w(i)
0
is abnormal (Lemma 7.27) and if w0 ∈ w(i) then Piw = Piw .
Therefore M 0 is a non-trivial probabilistic model.
Step 7
And now we are ready for our final proposition:
4. COMPLETENESS
65
Proposition 7.30. ∀w ∈ W : (M, w) φ iff (M 0 , w) φ
Proof. We will proceed by induction on the structure of φ.
The only interesting case is that of Biφ ψ.
(⇒)
Assume that for w ∈ W we have that: w M Biφ ψ.
w
Then bestw
i (||φ|| ∩ Wi ) ⊆ ||ψ||.
Now we move to our probabilistic model M 0 .
If ||φ|| ∩ w(i) = ∅, then ||φ|| is abnormal w.r.t. Piw , in which case w M 0 Biφ ψ.
6 ∅,
If ||φ||∩w(i) 6= ∅, consider S ⊆ w(i) the least r-stable set w.r.t. Piw such that S ∩||φ|| =
i.e. S ⊆ S 0 , ∀S 0 : r-stable sets w.r.t. Piw such that S 0 ∩ ||φ|| =
6 ∅.
Pick now x ∈ S ∩ ||φ||.
By the definition of Piw , we have that:
S ∩ ||φ|| = bestw
i (||φ|| ∩ w(i)) ⊆ ||ψ||.
Therefore ∅ =
6 S ∩ ||φ|| ⊆ ||ψ||, with S : r-stable set w.r.t. Piw .
Hence w M 0 Biφ ψ.
(⇐)
Assume that for some w ∈ W we have that: w M 0 Biφ ψ.
Now if ||φ|| is abnormal w.r.t. Piw , then by the definition of Piw , we have that: ||φ||∩S = ∅
for all S r-stable sets w.r.t. Piw . This in turn implies that bestw
i (||φ||) = ∅ and hence
φ
w M Bi ψ.
Now if ||φ|| is normal, then we have that ∃ an r-stable set S w.r.t. Piw such that:
∅=
6 S ∩ ||φ|| ⊆ ||ψ||.
Pick now the least r-stable set S 0 w.r.t. Piw such that ∅ =
6 S 0 ∩ ||φ||.
0
0
Then S ⊆ S hence S ∩ ||φ|| ⊆ S ∩ ||φ|| ⊆ ||ψ||.
Now, by the construction of the model M 0 we get that: if x ∈ S 0 ∩ ||φ||, then x ∈
w
bestw
i (||φ|| ∩ Wi ).
φ
w
Therefore bestw
i (||φ|| ∩ Wi ) ⊆ ||ψ|| and thus w M Bi ψ.
And we have now completed the proof of Proposition 7.21.
And finally:
Proposition 7.31. BRSID is a complete axiomatization w.r.t. non-trivial probabilistic frames.
Proof. Combine Propositions 7.21 and 7.9.
CHAPTER 8
Safe Belief, Certainty operators and the notion of quasi-stability
Now that we have established a logic for conditional belief based on the notion of
r-stability we are ready to proceed with our next goal: a logic with a language able to
express statements such as “agent i has an ri -stable belief that φ”.
The straightforward way of doing that, would be to introduce an operator Sbri . However, such an operator would not satisfy the K axiom. Therefore, we need another way
of defining r-stable beliefs. To do so, we adopted the following more fundamental modalities: : “safe belief” ([4]) and C: certainty. The idea comes from Baltag and Smets’
papers, in which they provide a definition of conditional belief in terms of safe belief ()
and their notion of knowledge (K): BiP Q = K̃i P → K̃i (P ∧i (P → Q)). In their setting,
is an S4 modality and K an S5. Now C for us would imply something stronger than
probability 1 but nevertheless weaker than absolute truth. For a set A in our algebra
CA would imply that A is a priori. Therefore, while Ac would be abnormal and hence
P (Ac |W ) = 0, that would not necessarily mean that Ac = ∅. On the other hand, will
be neither truthful nor negatively introspective in our probabilistic setting.
We will follow Baltag and Smets’ work in [6] and we will define conditional belief in
terms of and C. We will do so in ω-stable conditional probability spaces (Definition
4.16), when we have countably many stable sets. Sadly, we will still be unable to define
the notion of r-stability syntactically in terms of these operators. In fact, as we will
discuss at the end of this chapter, after we have presented a formal account of our theory,
in order to define the operator Sb(φ) (φ is an r-stable belief), we need the stronger S5
modality: K, instead of our C. We decided to tackle this problem by defining a more
general notion of stability, that of quasi-stability. This notion is based on the same idea
the quasi-subset relation was based on in chapter 4.
This chapter will be divided in 3 sections.
First, we will introduce the and C operators. We will characterize sets of the form
A for A ∈ F and prove some important properties about the operator.
Next, we will define the notion of quasi-stability. We will express it in terms of the and C operators and we will also show that in an ω-stable conditional probability space,
a set Σ is quasi-stable if it is the union of an r-stable set S and an abnormal set a:
Σ = S ∪ a.
Finally, we will move on to conditional belief. Once again we will work in ω-stable
conditional probability spaces and define conditional belief in terms of quasi-stable sets,
proving the most important result of this chapter, namely that conditional belief can be
fully expressed in terms of the and C operators, as long as we have countably many
r-stable sets.
67
68
8. SAFE BELIEF, CERTAINTY OPERATORS AND THE NOTION OF quasi-STABILITY
1. and C operators
Consider W a set of possible worlds, (W, F, P ) a conditional probability space and
fix some threshold r ∈ ( 12 , 1].
Now we will define two new operators:
Definition 8.1. For A ∈ F:
x ∈ A iff ∀E ∈ F(x ∈ E ⇒ B E A).
We read w ∈ A as saying: “at state w our agent safely believes that A”.
Definition 8.2. For A ∈ F we have that:
CA := B ¬A ⊥
We read CA as “A is an a priori set”. It is easy to see that the C operator is global
in the sense that either A is a priori on every state w ∈ W or A is not a priori anywhere
in W . However, C is unlike knowledge in the sense that it is not necessarily truthful:
Cφ → φ is not sound in our probabilistic semantics. As discussed above for a formula φ,
Cφ implies that P (||φ|||W ) = 1 and P (||φ||c |W ) = 0 but it is important to note that this
does not imply that ||φ||c = ∅, i.e. that ||φ|| = W .
Now we have the two following lemmas:
Lemma 8.3 (Monotonicity of the operator). For A, B ∈ F, if A ⊆ B, then A ⊆
B.
Proof. Take A, B ∈ F such that A ⊆ B.
Let x ∈ A.
Then we have that for all E ∈ F, if x ∈ E then B E A holds.
However, we have shown in chapter 5 (5.8) that if A ⊆ B and B E A holds we have that
B E B holds as well.
Therefore we get that ∀E ∈ F if x ∈ E then B E B holds.
Therefore x ∈ B.
Lemma 8.4. For a set A ∈ F we have that either A ⊆ A or that A is a priori.
Proof. Consider A ∈ F and assume that A 6⊆ A.
Then ∃x ∈ A such that x ∈ Ac .
c
But then we get that B A A, which entails that A is a priori.
Finally, the above lemma gives us the following interesting corollary:
Corollary 8.5. For all sets A ∈ F, we have that:
A ⊆q A
Now we will define r-stable beliefs in terms of the operator:
1. AND C OPERATORS
69
Proposition 8.6. S ∈ F is an r-stable set iff S 6= ∅ and S ⊆ S.
Proof. (⇒)
Consider S ∈ F an r-stable set.
Then S 6= ∅ by definition.
Now pick x ∈ S.
We need to show that x ∈ S, i.e. that B E S holds for all E ∈ F such that x ∈ E.
Pick E ∈ F such that x ∈ E.
Then we need to show that B E S holds.
• Case 1. E is abnormal. Then B E S holds.
• Case 2. E is normal. Then we have that x ∈ S ∩ E.
Therefore S ∩ E 6= ∅.
And we also have that S ∩ E ⊆ S with S an r-stable set, so indeed B E S holds.
(⇐)
Take S ∈ F such that S 6= ∅ with S ⊆ S.
We need to show that S is an r-stable set.
Pick E ∈ F such that S ∩ E 6= ∅.
Then since S ⊆ S we have that B E S holds.
This entails that ∃S 0 ∈ F such that S 0 is r-stable with S 0 ∩E a normal set and S 0 ∩E ⊆ S.
• Case 1. E is abnormal. Then P (S|E) = 1 ≥ r.
• Case 2. E is normal. Then P (S|E) ≥ P (S 0 ∩ E|E) = P (S 0 |E) ≥ r, since S 0 is
r-stable and S 0 ∩ E 6= ∅.
Therefore S is r-stable and our proof is complete.
The above is an important result that connects the operator directly with r-stability.
The following propositions characterize sets of the form A.
Proposition 8.7. Let A ∈ F. Then w ∈ A holds iff we have either
• A is a priori, or
• ∃S : r-stable set in F such that w ∈ S and S ⊆ A.
Proof. (⇒)
Suppose that w ∈ A but 6 ∃S : r-stable set such that w ∈ S and S ⊆ A.
Given this we want to show that A is a priori.
Call H the family H := {S ∈ F|S : r−stable and S ⊆ A}.
We have that H is non-empty, since we have that w ∈ W and w ∈ A, hence B W A
holds.
Therefore ∃S : r-stable set such that S ∩ W = S is normal and S ∩ W = S ⊆ A.
Hence this particular S is in H.
S
Now define D := H, the union of the elements of H.
Now ∀H ∈ H : H ⊆ A, hence D ⊆ A, therefore Ac ⊆ Dc .
Now we have assumed that w does not belong to any of the sets in H, since we have
assumed that 6 ∃S : r-stable set such that: w ∈ S and S ⊆ A.
Hence w does not belong in D as well. Therefore w ∈ Dc .
Now we want to show that Ac is abnormal.
To do this, it is enough to show that Dc is abnormal, by Property 3.12.
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8. SAFE BELIEF, CERTAINTY OPERATORS AND THE NOTION OF quasi-STABILITY
Claim: Dc is abnormal
c
Proof. Suppose Dc is normal. Then since w ∈ A and w ∈ Dc we get B D A.
Therefore ∃S 0 : r-stable set such that S 0 ∩ Dc is normal and S 0 ∩ Dc = S 0 − D ⊆ A.
However, we also have that S 0 ∩ D ⊆ A, since S 0 ∩ D ⊆ D and D ⊆ A.
Therefore: S 0 = (S 0 − D) ∪ (S 0 ∩ D) ⊆ A.
But then S 0 ∈ H. Hence S 0 ⊆ D, meaning that S 0 − D = ∅.
But this contradicts that S 0 − D is normal.
Hence Dc is indeed abnormal.
This entails that Ac is abnormal as well, as its subset and this gives us that A is a
priori, as desired.
(⇐)
Consider A ∈ F and x ∈ W .
We split our premise into two subcases:
• Case 1. A is a priori.
Then ∀E ∈ F we have: B E A.
Therefore: A = W .
• Case 2. A is contingent and ∃S : r-stable set such that x ∈ S and S ⊆ A.
Let E ∈ F be any set in the algebra such that x ∈ E.
Then E ∩ S ⊆ A.
Moreover, A is contingent and S ⊆ A. Therefore S is contingent.
Also, E ∩ S 6= ∅, since x ∈ E.
Now by Corollary 4.12 S ∩ E is normal.
Therefore B E A holds.
Hence x ∈ A.
Corollary 8.8. We have that
(a) if A ∈ F is a priori, then A = W .
(b) if A ∈ F is contingent, then A is the union of all the stable subsets of A.
Proposition 8.9. For A ∈ F, we have that if A is a priori then A = W .
Proof. Let A ∈ F such that A is a priori.
Then we have two cases:
• Case 1: A is a priori. Then by Corollary 8.8 we have that A = W .
• Case 2: A is contingent. Then Lemma 8.4 tells us that A ⊆ A.
But this together with A being a priori, implies that A is a priori. Contradiction.
2. quasi-STABLE SETS
71
Proposition 8.10. For all A ∈ F, we have that: A = A.
Proof. Pick A ∈ F. We have the two following cases:
• Case 1. A is a priori.
Then by Corollary 8.8 we have that A = W , which is also a priori, hence
A = W = W = A.
• Case 2. A is contingent.
Then by Corollary 8.8 A is the union of all stable subsets of A. Also, A ⊆ A
in this case, thus A is contingent (since A is), hence A is the union of all
stable subsets of [the union of all stable subsets of A].
But (using nestedness of contingent stable sets (Corollary 4.6)), this is just the
union of all stable subsets of A, i.e. A.
Finally, we have a result for sets of the form A.
Proposition 8.11. Consider (W ω , F ω , P ω ) to be an ω −stable conditional probability
space as in Definition 4.16.
Then the sets of the form A for A ∈ F ω are stable.
Proof. Pick A ∈ F ω . We have two cases:
• Case 1. If A is a priori, then A = W ω , in which case it is stable.
• Case 2. If A is contingent, by Corollary
S 8.8 we have that A is the union of all
the stable subsets of A, i.e. A = Sk with Sk contingent (as subsets of A)
k
stable sets. By the definition of the ω − stable conditional probability space, we
get that
S these stable sets Sk are countably many. Now by Property 4.8 we have
that Sk is a stable set. Therefore A is indeed a stable set.
k
Notice that the set
S
Sk is stable, only because we have countably many stable sets.
k
In more general (not necessarily ω-stable) conditional probability spaces, Proposition 8.11
does not hold.
2. quasi-stable sets
In this section we will define a new notion of stability, called: quasi-stability.
Let (W, F, P ) be a conditional probability space.
Definition 8.12. We will say that a set Σ ∈ F is quasi-stable and write: QSb(Σ)
iff Σ 6= ∅ and ∃a abnormal set such that Σ ⊆ Σ ∪ a.
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8. SAFE BELIEF, CERTAINTY OPERATORS AND THE NOTION OF quasi-STABILITY
Notice that the above definition also entails that a set Σ ∈ F is quasi-stable if and
only if Σ 6= ∅ and Σ ⊆q Σ, with ⊆q the quasi-subset relation defined in chapter 4
(Definition 4.3).
The following proposition connects quasi-stable and r-stable sets.
Proposition 8.13. Let (W ω , F ω , P ω ) be an ω-stable conditional probability space as
in Definition 4.16.
Now let Σ ∈ F ω . Then Σ is quasi-stable iff there exists a stable set S ∈ F ω and an
abnormal set a ∈ F ω such that Σ = S ∪ a.
Proof. (⇒) Let Σ ∈ F ω a quasi-stable set.
We have the following cases:
• Case 1. Σ is a priori. Then Σ = Σ∪∅, with Σ stable (as a priori) and ∅ abnormal.
• Case 2. Σ is contingent.
S Then by Corollary 8.8 Σ is the union of all the stable
subsets of Σ: Σ = Sn with Sn stable sets. Since we are in an ω-stable space,
n
we
S have that these Sn sets are countable. Now by Property 4.8 we have that
Sn is stable, thus Σ is stable.
n
Thus we take S = Σ and this direction is complete.
(⇐)
Let S ∈ F stable, i.e. S 6= ∅ and S ⊆ S.
Let a ∈ F abnormal. We have S ∪ a ⊆ S ∪ a and S ∪ a 6= ∅.
Hence by Definition 8.12 S ∪ a is quasi-stable.
3. Conditional Belief: quasi-stability and , C operators
In the final section of this chapter, we are going to define conditional belief in terms
of the and C operators, analogously to what Baltag and Smets did in [6].
In this section we will be working in ω-stable conditional probability spaces as in Definition
4.16. In these spaces we have countably many contingent stable sets.
For notational simplicity we take W ω = W , F ω = F and P ω = P .
First we define conditional beliefs in terms of quasi-stable sets.
Proposition 8.14. For E, H ∈ F, we have: B E H iff either E is abnormal or ∃Σ :
quasi − stable set such that Σ ∩ E is normal and Σ ∩ E ⊆ H.
Proof. (⇒) Assume that B E H holds for E, H ∈ F.
Then ∃S stable set such that S ∩E is normal and S ∩E ⊆ H. Now since ∅ is an abnormal
set, we take Σ = S ∪ ∅ = S and we are done.
(⇐) Take E, H ∈ F such that E is normal (if E is abnormal then it is trivial) and
assume that there exists a set Σ ∈ F such that Σ is a quasi-stable set with Σ ∩ E a
3. CONDITIONAL BELIEF: quasi-STABILITY AND , C OPERATORS
73
normal set and Σ ∩ E ⊆ H.
Then by Proposition 8.13 ∃S ∈ F such that S is stable and Σ = S ∪ a with a ∈ F an
abnormal set. We have the following two cases:
• Case 1. S is a priori.
Then if E is normal we get that S ∩ E is normal.
We also have that S ∩ E ⊆ Σ ∩ E ⊆ H and we are done.
• Case 2. S is contingent.
Then Σ ∩ E = (S ∩ E) ∪ (a ∩ E).
But Σ ∩ E is a normal set and a ∩ E an abnormal set, therefore S ∩ E should be
a normal set as well, since otherwise we would have that Σ ∩ E is abnormal by
Property 3.13.
Having also that S ∩ E ⊆ H we derive that B H E holds.
Hence we have established a new definition for conditional belief in terms of quasistable sets. However, this result holds only in ω-stable spaces.
And now we will move on to the central point of this section, which is the definition
of conditional belief in terms of the and C operators:
(use C̃(E) for ¬C(¬E) = ¬C(E c ))
Proposition 8.15. B E H iff C̃E ⇒ (C̃(E ∧ (E → H)))
Proof. First notice that C̃E = ¬C¬E essentially means that E is normal.
(⇒)
Assume that B E H holds for some E, H ∈ F.
We have two cases:
• Case 1. E is abnormal. Then the right hand side implication holds trivially.
• Case 2. E is normal.
We want to show that E ∩ (E → H) is normal as well.
Since B E H holds, we get that:
∃S : r-stable set such that S ∩ E is normal and S ∩ E ⊆ H.
Since S is stable we have that S ⊆ S.
Moreover, we have that S ⊆ (S ∩ E) ∪ (S ∩ E c ).
Now S ∩ E ⊆ H ⊆ E c ∪ H.
Also S ∩ E c ⊆ E c .
Therefore S ⊆ (S ∩ E) ∪ (S ∩ E c ) ⊆ (E c ∪ H) = (E → H).
Therefore S ⊆ E → H.
Now by 8.3 we have that is a monotonic operator.
Therefore S ⊆ (E → H).
Hence E ∩ S ⊆ E ∩ S ⊆ E ∩ (E → H).
Now E ∩ S is a normal set, otherwise by 4.11 we would have that S is a priori,
in which case we would have by 4.13 that E itself is abnormal, contradicting our
assumption.
Therefore E ∩ (E → H) is a superset of a normal set and therefore by Property
3.11 we get that E ∩ (E → H) is a normal set as well.
(⇐)
Assume that for some sets E, H ∈ F we have that C̃E ⇒ C̃(E ∩ (E → H)) holds.
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8. SAFE BELIEF, CERTAINTY OPERATORS AND THE NOTION OF quasi-STABILITY
As written above this means that if E is normal then E ∩ (E → H) is normal as well.
We want to show that B E H holds.
We have two cases:
• Case 1. E is abnormal. Then by Corollary 5.2 B E H holds.
• Case 2. E is normal.
Then we have the two following subcases:
– Subcase 1. (E → H) is not a priori.
Then (E → H) ⊆ (E → H), by Lemma 8.4.
Therefore E ∩ (E → H) ⊆ H, since (E → H) = E c ∪ H.
Now by Proposition 8.11, the set (E → H) is stable.
Moreover we know that E ∩ (E → H) is normal by our assumptions.
Hence combining these two we get that B E H holds.
– Subcase 2. (E → H) is a priori.
Then (E → H)c = E − H is abnormal.
However we have that E is normal.
Hence E ∩H is normal, otherwise E = (E −H)∪(E ∩H) would be abnormal
contradicting our assumption.
Now since (E → H) is a priori, it is also stable.
Therefore we have that: E ∩ (E → H) = E ∩ H ⊆ H, with (E → H) stable
and E ∩ (E → H) a normal set, since E ∩ H is normal.
Therefore B E H holds.
We saw that S is stable if and only if S 6= ∅ and S ⊆ S.
This is equivalent to saying that S is stable if and only if S 6= ∅ and (S ∩ (S)c ) 6= ∅. To
express this in our language, we need an operator K such that Kφ → φ holds. In that
case we could define an operator Sb for stable beliefs as:
Sb(S) ⇔ ¬K¬S ∧ K(S → S).
Then ¬K¬S tells us that S 6= ∅ and that S c ∪ S holds, i.e. that (S ∩ (S)c ) 6= ∅, i.e.
that S ⊆ S.
However, the closest we can get to this K operator is C. CA now tells us that A is
a priori. Now if we try to define stable sets in terms of the C operator, we run into
problems:
Sb(S) ⇔ ¬C¬S ∧ C(S → S).
Now the first part: ¬C¬S tells us that S is normal. As we explained in chapter 4 this is in
fact equivalent to saying that S is non-empty when it comes to r-stability. The problem
however comes in the second part of the conjunction. This is because C(S → S) tells
us that the set S c ∪ S is a priori, i.e. that the set S ∩ (S)c is abnormal, i.e. that there
exists an abnormal set a such that S ⊆ S ∪ a, i.e. that S ⊆q S. But this is not enough
to conclude that S is stable. This observation motivates the concept of quasi-stability.
As we saw in Proposition 8.14, when in ω-stable conditional probability spaces defining
conditional beliefs in terms of quasi-stable sets is equivalent to our old definition that was
in terms of stable sets. Therefore we can express conditional beliefs using quasi-stable
sets. However, this is possible only in spaces that we have countably many stable sets.
CHAPTER 9
Conclusion
1. Overview
The goal of this thesis was twofold. First, to extend Leitgeb’s theory of stable beliefs
into non-classical probability spaces, where conditioning on events with measure 0 is
defined. Furthermore, to develop a formal language in order to express the notion of
conditional belief.
We first gave a brief presentation of Leitgeb’s stability theory as found in his papers
[35], [34]. We argued that his theory is a path between the materially wrong proposal
that equates belief with probability 1 and the logically wrong Lockean Thesis (or any
version of it). We believe that Leitgeb’s formalization of belief is more intuitive than the
current alternatives and at the same time it comes without a cost, since we maintain the
logical closure of belief without running into the lottery paradox. However, his classical
probability setting is not enough to do belief revision, because conditioning on events of
measure 0 can not be defined.
Therefore, our first goal was to extend Leitgeb’s ideas into a non-classical probability
space. According to Halpern ([29]) the three most popular approaches of dealing with
conditioning on sets of measure 0 are: conditional probability spaces, lexicographic probability spaces and nonstandard probability spaces. We chose to develop our notion of
r-stable sets using conditional probability spaces ([12], [13], [29]) and more specifically in
Van Fraassen’s setting ([25], [3], [21]).
We used our two-place probability functions, to define the notion of r-stability similarly to Leitgeb. A set is r-stable if its probability remains above the threshold r given
the occurrence of any event consistent with it. We also proved that r-stable sets are
nested w.r.t. the ⊆q relation and that there is no infinitely descending chain of contingent r-stable sets.
The next step was to define conditional belief in terms of r-stable sets: agent i believes
H given E if and only if either E is abnormal, or there is an r-stable set S such that
S ∩E 6= ∅ and S ∩E ⊆ H. This definition comes in accordance with what Leitgeb calls the
“Humean conception of belief”, ([35, p. 33]) that “it is rational to believe a proposition
in case it is rational to have a stably high degree of belief in it”. Moreover, if E (the
evidence set) is abnormal, we followed Van Fraassen’s idea that the agent is so confused
that does not know what to believe anymore and hence he believes everything. Finally,
we showed that our conditional belief is closed under conjunction and is consistent w.r.t.
normal sets.
Our next goal was to develop a formal language to express statements such as “agent
i believes H given E”.
75
76
9. CONCLUSION
In chapter 6 we defined the structures called probabilistic frames; the structures that
gave us the semantics for our logic of conditional belief in chapter 7. They are essentially
the multi-agent version of the conditional probability spaces we defined in chapter 3 for
Ag a finite set of agents. Now a probabilistic frame is a structure in which the set of
possible worlds W is divided into partitions Πi : one for each agent i ∈ Ag. Therefore, for
each world w ∈ W we have an information cell w(i) for each i, induced by the partition.
Moreover, we have a function r that assigns a number ri ∈ ( 21 , 1] to each of our agents;
their threshold. Finally, we defined a function Pi that assigns a two-place probability
function Piw to each agent i at each state w. Therefore, we are now talking about ri stable sets w.r.t. Piw . At the end of chapter 6, we made a comparison of our notion of
r-stable and conditional belief with Battigalli and Siniscalchi’s. It appeared that B-S’s
“strong belief” can be expressed in our terms and is essentially the 1-stable belief (r-stable
belief for threshold r = 1).
In chapter 7, we presented our logic of r-stable conditional beliefs. In the first section
we presented our language, that of epistemic logic with the addition of the Biφ ψ operator
expressing “agent’s i belief in ψ given φ” and our semantics, the probabilistic models.
In section 2, we presented our axiom system, which is similar to Board’s in his logic
of conditional beliefs ([18]). In section 3 we proved that our axioms are sound in our
probabilistic semantics and in section 4 we proved the completeness of our axiom system
w.r.t. our probabilistic models.
Our next goal was to introduce a formal language that would allow us to express
statements such as “agent i has a stable belief in agent’s j rationality”. The straightforward way to achieve this, would be to introduce an operator Sbri (φ) stating: “φ is an
r-stable belief for agent i”. However, such an operator would not satisfy the K axiom.
Therefore, we had to go another way: express r-stable beliefs in terms of other operators
that do satisfy the K axiom. We chose Baltag and Smets’ modality ([4]) for safe belief
and the modality C for certainty.
However, these operators were still unable to capture our notion of r-stable sets. This
is because (as we argue in the last part of chapter 8) and C are not truthful in our
setting. Therefore, we defined a new notion of stability: that of quasi-stability. This
notion is minimally different from r-stability in the sense that it is more general. We
showed that in a conditional probability space with countably many stable sets, a quasistable set is the union of an r-stable set with an abnormal set. Therefore, a quasi-stable
belief has an abnormal part. We also showed that in such spaces, our notion of conditional
belief can be equally expressed in terms of quasi-stable sets.
2. Future Work
The first step after this thesis, is to develop the logic of certainty and safe belief based
on the modalities: and C. Let us call this logic QSBL.
Below, we will present our ideas about the language, semantics and a possible axiomatization of QSBL.
The language of QSBL, LQSBL can be defined as follows:
Definition 9.1. Consider a set of atomic sentences At. Our language LQSBL is a
set of formulas φ of QSBL and is defined recursively:
φ ::= p|¬φ|φ ∧ φ|φ|Cφ
2. FUTURE WORK
77
for p ∈ At.
Once again, we have the language of epistemic logic augmented by adding and C
operators. As it has been mentioned above φ is read as: “agent safely believes that φ”
and Cφ is read as: “agent is certain of φ”.
We use the standard abbreviations: φ ∨ ψ for ¬(¬φ ∧ ¬ψ), φ → ψ for ¬φ ∨ ψ and
φ ↔ ψ for (φ → ψ) ∧ (ψ → φ). Moreover, define ⊥ = p ∧ ¬p for p ∈ At and > = ¬⊥.
Finally abbreviate ¬C¬φ as C̃φ.
Now define the following operators:
Definition 9.2. For φ, ψ formulas in LQSB , we define:
B φ ψ = C̃φ → C̃(φ ∧ (φ → ψ))
As mentioned above conditional belief is defined in terms of C and . For the validity
of this definition, look at Proposition 8.15.
Definition 9.3. For φ a formula in LQSB , we define:
QSb(φ) = C̃φ ∧ C(φ → φ)
Here we defined the notion of quasi-stability in terms of C and . This definition is
motivated by our discussion at the last part of chapter 8.
Now for our semantics first consider a number r ∈ ( 12 , 1].
Now one should take care to note that the above definitions only hold in ω-stable conditional probability spaces. Therefore our probabilistic frames and models should be in
ω-stable spaces, in which we have countably many stable sets.
With this in mind, our semantics are given by the structure
M = (W, F, P, || · ||)
with
• (W, F, P ) is a non-trivial ω-stable conditional probability space and
• || · || : LQSBL → F a valuation.
M will be called a single agent ω-probabilistic model.13
As before, we require that for p ∈ At: ||p|| ∈ F.
We define truth as follows:
Definition 9.4. Consider M = (W, F, P, || · ||) a single agent probabilistic model,
w ∈ W and p ∈ At. The relation between pairs (M, w) and formulas φ ∈ LQSBL is
defined as follows:
• (M, w) φ iff w ∈ ||p||,
• (M, w) ¬φ iff w ∈ ||φ||c ,
• (M, w) φ ∧ ψ iff w ∈ ||φ|| ∩ ||ψ||,
• (M, w) φ iff ∀E ∈ F(w ∈ E ⇒ B E ||φ||),
• (M, w) Cφ iff ||φ|| ∈ F is a priori.
13Notice
that the single agent probabilistic model is essentially the single-agent restriction of our
probabilistic model.
78
9. CONCLUSION
Finally, one should take care to show that || · || is well-defined and in F.
Notice that since we are in ω-stable conditional probability spaces, sets of the form
A for some A ∈ F will be measurable. This is because if A ∈ F is a priori, then by
Corollary 8.8 A = W and therefore A ∈ F. If A is contingent, then by Corollary 8.8
we have that A is the union of all the stable subsets of A. However, these are countably
many (ω-stable conditional probability spaces) and therefore their union is in the algebra.
Hence A is measurable.
Now here is an axiom system for this logic. Consider φ, ψ ∈ LQSBL .
Our axioms and inference rules are the following, along with MP:
K
PI
NI
K
PI
Taut
Nec
for C
for C
for C
for for (a)
(b)
(c)
(d)
true
from φ infer Cφ and φ
C(φ → ψ) ⇒ (Cφ → Cψ)
Cφ → CCφ
¬Cφ → C¬Cφ
(φ → ψ) ⇒ (φ → ψ)
φ → φ
Cφ → φ
¬⊥
φ ∧ ¬Cφ → φ
C(φ → ψ) ∨ C(ψ → φ)
The first important observation is that neither of and C is truthful. The axioms
φ → φ and Cφ → φ can not be sound in our probabilistic semantics.
Furthermore, is not negatively introspective. Now Cφ → φ says that if φ is a
priori, then φ is a safe belief. In other words, this axiom tells us that the notion of a
priori is stronger than that of safe belief. ¬⊥ says that contradiction can not be safely
believed.
Finally, φ ∧ ¬Cφ → φ is as close we can get to factivity. This axiom says that if φ
is safely believed, then either φ is the case, or φ is a priori. Intuitively, this axiom says
that an agent can be “wrong” about his safe beliefs, only if what he safely believes is an
a priori set. In other words, safe belief looses its “credibility” when it coincides with one
of our agent’s a priori convictions (certainties).
Now the next step is proving soundness and completeness w.r.t. the single agent
probabilistic models.
At this point, we would like to write some comments about the completeness proof.
Completeness could be established in a way similar to what we did in section 7. First,
we could use canonical models as in [15] and prove completeness w.r.t. finite partial
plausibility models:
Definition 9.5. The structure (W, S, ≤, || · ||) such that:
• W is a finite set of possible worlds,
• S ⊆ W,
• ≤ a plausibility ordering on S such that:
2. FUTURE WORK
79
– ≤ is complete and transitive on S,
– ≤ is well-founded
• || · || a function assigning formulas of our language to subsets of W ,
will be called a finite partial plausibility model.
Afterwards, we can “probabilize” a finite partial plausibility model in a way similar
to what we did in section 7 and obtain a single agent probabilistic model.
Finally, we prove a truth preserving lemma between finite partial plausibility models
and single agent ω-probabilistic models and we derive completeness w.r.t. our probabilistic semantics.
Now the next step is the multi-agent version of the logic of certainty and safe belief
(QSBL).
The language of this logic can be
φ ::= p|¬φ|φ ∧ φ|φi φ|Ci φ
for i ∈ Ag with Ag a finite set of agents.
The semantics will be given by our probabilistic frames as defined in chapter 6. However, we will be referring to ω-stable conditional probability spaces.
Once we have the multi-agent logic of certainty and safe beliefs, the next step is to
introduce the notions of Mutual and Common quasi-stable beliefs in φ: EQSb(φ) and
CQSb(φ) respectively.
Now EQSb(φ) is defined as usual:
V
EQSb(φ) :=
QSbi (φ)
i∈Ag
and the usual axioms for CQSb(φ) are:
• CQSb(φ) ⇒ EQSb(φ ∧ CQSb(φ))
• CQSb(φ → EQSb(φ)) ⇒ (EQSb(φ) → CQSb(φ))
At this point there are two possible directions one could take. Either introduce a logic
of Common quasi-stable beliefs along with the necessary soundness and completeness
proofs, or simply define these notions and apply them.
One possible application would be the characterization of the epistemic conditions of
backward/forward induction.
We have already mentioned that Battigalli and Siniscalchi use their notion of strong
belief to characterize backward induction.
Baltag, Smets and Zvesper do something analogous in their paper [10]. They introduce
a new notion of rationality: dynamic rationality and use the notion of “stable belief”,
belief that is preserved during the play of the game, to provide a characterization of the
epistemic conditions for backward induction: dynamic rationality and common knowledge
of stable belief in rationality.
Here we will give a very brief and informal presentation of their work, only to show
that there is a direct connection to our theory.
First, concerning their notion of dynamic rationality.
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9. CONCLUSION
On one hand, this notion assesses the rationality of a player’s move at a node w.r.t.
the beliefs held when this node is reached ([10, p. 304]). On the other hand, this notion of
rationality also incorporates the epistemic limitation to rationality: the rationality of an
agent’s move only makes sense when that move is not already known to her ([10, p. 304]).
They argue that their notion of rationality is future-oriented in the sense that at any
stage of the game, the “dynamic rationality” of an agent depends only on her current
and future moves. Therefore, a player can be rational now, even if he has made irrational
moves before ([10, p. 305]). Suppose now that player i behaves irrationally at some node.
Then the rest of the players do learn that i is irrational at this moment, but this may as
well be forgotten as the game continues. This means that a previously irrational player
can become rational after her wrong move. The intuition is that she might choose the
right moves for all the decisions that she can still make ([10, p. 305]).
Hence, the meaning of “rationality” changes in time, due to the change of beliefs and
of the known set of options ([10, p. 305]).
Their main argument is that the rationality of a player is an epistemic-doxastic concept, so it should be affected by any changes of the information of the player.
The main theorem of their paper is the following:
Common knowledge of (the game structure, of “open future” and of) stable common
belief 14 in dynamic rationality entails common belief in the backward induction outcome
([10, p. 306]).
Now the “open future” assumption essentially is that players have no non-trivial
“hard”15 information about the outcomes of the game ([10, p. 329]).
The notion of “stable belief” corresponds to Baltag and Smets’ notion of safe belief
([4]): an analogy of the operator we used in our chapters 8, 9. However, as it has
been mentioned quite a few times before, there is a big difference (amongst others): our
version is not truthful.
The first question that comes up at this point, is what would happen if we used our
quasi − stale belief in the place of their “stable belief”. So suppose that we assumed that
we have common knowledge of the game, of “open future” and of quasi-stable common
belief in dynamic rationality. Would we get common belief in the backward induction
outcome? Moreover, to what extent would the result depend on the selection of the
threshold r?
The next point, is that B-S’s work in [13] (briefly presented in chapter 2) is really
close to what Baltag, Smets and Zvesper did. However, B-S’s notion of rationality is
only “partially-dynamic”: it requires the agents to make rational choices at all the nodes,
including the ones that have already been bypassed ([10, p. 329]). Therefore, this implies
that one irrational move is enough to break the common belief in rationality.
This is why B-S use the assumption of the complete-type structure (Definition 2.4) in
order to characterize forward induction by assuming common strong belief in rationality
(with their notion of strong belief as presented above). A complete-type structure contains
every possible epistemic-doxastic type of each player. This assumption demands that
players consider all probabilistic assignments as epistemically possible ([10, p. 330]).
14As
they state: “common” here is not necessary, since common knowledge that everybody has a
stable belief in P is the same as common knowledge of common safe belief in P ([10, p. 306]).
15
“hard” as absolutely “indefeasible”, unrevisable information as in [41]
3. CONCLUSION
81
Therefore, the next interesting question is whether we can assume common quasistable belief in dynamic rationality and characterize the epistemic conditions of forward
induction in a way similar to B-S, only by “loosening” their really strict assumption
of complete-type structures. Once again, how would the choice of the threshold r be
relevant?
3. Conclusion
Our venture in this thesis was mainly — if not exclusively — technical. With many
results in probability spaces and one long completeness proof in chapter 7, one could
accuse us of lacking convincing arguments as to why the theory of stable beliefs is worth
dealing with in the first place.
However, arguing in favor of Leitgeb’s theory was outside of the scope of our work.
Instead, being convinced that Leitgeb’s theory offers us both a philosophically intuitive
and logically acceptable (or better yet not counter-intuitive and not unacceptable) quantitative probabilistic representation of belief, we aimed to provide a common conceptual
framework that unifies Leitgeb’s notion of stable belief, B-S’s notion of strong belief and
the notions of a priori, abnormal and conditional belief studied by Van Fraassen and
Arló-Costa.
As we mentioned in the beginning of this thesis, there are turbulent waters between
logic and probability. In [34, p. 1388] Leitgeb writes: “For now, we hope to have laid the
foundations of what will hopefully become a valuable contribution to the “peace project”
between logic and probability theory”.
In turn, we hope that this thesis has strengthened these foundations.
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